series representations and asymptotic expansions of

SERIES REPRESENTATIONS AND ASYMPTOTIC EXPANSIONS OF EXTENDED GENERALIZED HYPERGEOMETRIC FUNCTION Anatoly A. Kilbasa , R.K. Saxenab , Megumi Saigoc and Juan J. Trujillod a

Department of Mathematics and Mechanics, Belarusian State University Nezavisimosti ave 4, Minsk, 220030 Belarus b Department of Mathematics and Statistics, Jai Narain Vyas University Jodhpur 342011 India c Department of Applied Mathematics, Fukuoka University Fukuoka 814-0180 Japan d Departamento de Análisis Matemático, Universidad de La Laguna La Laguna-Tenerife 38271 Spain

The paper is devoted to the study of a certain function p Fq [z] ≡ p Fq [a1 , · · · , ap ; b1 , · · · , bq ; z], with complex z 6= 0 and complex parameters ai (i = 1, · · · p) and bj (j = 1, · · · , q), represented by the Mellin–Barnes integral. Such a function is an extension of the classical generalized hypergeometric function p Fq [a1 , · · · , ap ; b1 , · · · , bq ; z] defined for all complex z ∈ C when p < q + 1 and for |z| < 1 when p = q + 1. Conditions are given for the existence of p Fq [z] and of its representations as the Meijer G-function and the H-function. Such an approach allows us to take a meaning to the function p Fq [z] for all ranges of parameters when p < < q + 1, p = q + 1 and p > q + 1. The series representations and the asymptotic expansions of p Fq [z] at infinity and at the origin are established. Special cases are considered. KEY WORDS: extended generalized hypergeometric function, Mellin–Barnes integral, H-function, Meijer G-function, generalized hypergeometric series, Gauss hypergeometric function, series representations, asymptotic expansions MSC (2000): 33C60, 33C20, 33C05, 41A60

1 Introduction The classical generalized hypergeometric function p Fq [a1 , · · · , ap ; b1 , · · · , bq ; z] is defined by the generalized hypergeometric series [1, 4.1(1)] p Y

(ai )k ∞ X zk i=1 , p Fq [z] ≡ p Fq [a1 , · · · , ap ; b1 , · · · , bq ; z] = q Y k! k=0 (bj )k

(1.1)

j=1

where z ∈ C and p, q ∈ N0 = {0, 1, 2 · · · } and ai , bj ∈ C (i = 1, · · · , p; j = 1, · · · , q). Here C is the set of complex numbers, an empty product in (1.1), if it occurs, is taken to

44

EXTENDED GENERALIZED HYPERGEOMETRIC FUNCTION

be one, and (d)k (d ∈ C, k ∈ N0 ) is the Pochhammer symbol [1, 1.21(5)]: (d)0 = 1,

(d)k = d(d + 1) · · · (d + k − 1)

(k ∈ N = {1, 2, · · · }),

(1.2)

The generalized hypergeometric function includes many elementary and special functions. In particular, when p = q = 0, (1.1) coincides with the exponential function z 0 F0 [−; −; z] = e ≡

∞ X zk k=0

k!

.

(1.3)

If p = 2 and q = 1, then (1.1) gives the Gauss hypergeometric function [1, 2.1(2)] 2 F1 [a1 , a2 ; b1 ; z] =

∞ X (a1 )k (a2 )k z k , (b1 )k k!

(1.4)

k=0

while for p = 3 and q = 2 (1.1) yields the hypergeometric function ∞ X (a1 )k (a2 )k (a3 )k z k . 3 F2 [a1 , a2 , a3 ; b1 , b2 ; z] = (b1 )k (b2 )k k!

(1.5)

k=0

It is known [1, Section 4.1] that the generalized hypergeometric function defined by the series in (1.1), is absolutely convergent for all finite z ∈ C if p < q + 1 and for |z| < 1 if p = q + 1. In particular, the series in (1.4) and (1.5) are absolutely convergent for |z| < 1. It is also known [1, 5.6(1)], [6, 7.2.3.12, 7.2.3.13] that if bj 6= − l (j = 1, · · · , q; l ∈ ∈ N0 ), then the generalized hypergeometric function (1.1) can be represented in terms of the Mellin–Barnes integral [1, Section 1.19] of the form p Fq [z] = p Fq [z], where p Fq [z]≡ p Fq [a1 , · · · q Y j=1 = p Y i=1

Γ(bj ) Γ(ai )

, ap ; b1 , · · · , bq ; z]

1 2πi

Z Γ(s)

p Y

Γ(ai − s)

i=1 q Y L

(−z)−s ds (z6=0)

(1.6)

Γ(bj − s)

j=1

with the special chosen contour L. Such a formula leads to the representation of (1.1) as the Meijer G-function (see Section 3). In particular, the hypergeometric functions (1.4) and (1.5) have for 0 < |z| < 1 the representations 2 F1 [a1 , a2 ; b1 ; z] = = 2 F1 [a1 , a2 ; b1 ; z] and 3 F2 [a1 , a2 , a3 ; b1 , b2 ; z] = 3 F2 [a1 , a2 , a3 ; b1 , b2 ; z], where Z 1 Γ(b1 ) Γ(s)Γ(a1 − s)Γ(a2 − s) (−z)−s ds (1.7) 2 F1 [a1 , a2 ; b1 ; z] = Γ(a1 )Γ(a2 ) 2πi Γ(b1 − s) L

A. KILBAS, R. SAXENA, M. SAIGO, J. TRUJILLO

45

with b1 6= − l (l ∈ N0 ), and 3 F2 [a1 , a2 , a3 ; b1 , b2 ; z]

=

Γ(b1 )Γ(b2 ) 1 Γ(a1 )Γ(a2 )Γ(a3 ) 2πi

Z L

Γ(s)Γ(a1 − s)Γ(a2 − s)Γ(a3 − s) (−z)−s ds Γ(b1 − s)Γ(b2 − s)

(1.8)

with bj 6= − l (j = 1, 2; l ∈ N0 ), respectively. In this paper we use the above representation (1.6) as the definition of the extended generalized hypergeometric function p Fq [z]. Choosing the contour of integration L, we give conditions for the existence of such a function for all finite z6=0 and for all ranges of parameters in respective cases p < q + 1, p = q + + 1 and p > q + 1. We deduce the representations for p Fq [z] as the Meijer Gfunction and as the H-function. Using the residue theory and asymptotic properties of the H-function, we establish series representations for p Fq [z] and deduce its asymptotic expansions at zero and infinity. In particular, the corresponding results are given for 2 F1 [a1 , a2 ; b1 ; z], 2 F2 [a1 , a2 ; b1 , b2 ; z], 3 F2 [a1 , a2 , a3 ; b1 , b2 ; z] and 3 F3 [a1 , a2 , a3 ; b1 , b2 , b3 ; z]. The above approach allows us to give a meaning to the function p Fq [z] for all ranges of parameters when p < q + 1, p = q + 1 and p > q + 1. In this connection we note that there is another approach, based on the MacRobert‘s E-function, to give a meaning in the case p > q + 1 for the classical hypergeometric function p Fq [a1 , · · · , ap ; b1 , · · · , bq ; z], defined by (1.1) for p 5 q +1; see, for example, [1, Section 5.2]. The paper is organized as follows. Section 2 contains results giving conditions for the existence of the extended generalized hypergeometric function p Fq [z] defined by the Mellin–Barnes integral (1.6). A representation of this function as the Meijer Gfunction is given in Section 3. The series representations and asymptotic expansions of such a generalized hypergeometric function are established in Section 4 and 5, respectively. Similar results to the special cases of p Fq [z] with p = 2, q = 1; p = q = 2; p = 3, q = 2 and p = q = 3 are presented in Section 6. We note that the classical generalized hypergeometric function (1.1) arises in many applied problems; for example, see [4]. The results presented in our paper can also be useful in such applied problems. In particular, the obtained explicit series representations and the explicit asymptotic expansions of the considered extended generalized hypergeometric functions can be used for the approximation of these functions. See [3] in this connection. We also think that the results obtained could be useful to consider asymptotic behavior of the functions considered when some of their parameters are large. Such results for the Gauss hypergeometric function 2 F1 [a1 , a2 ; b1 ; z] are well developed; see, for example, [11].

2 Extended generalized hypergeometric functions as the Mellin–Barnes integrals In this section we give conditions for the existence of the extended generalized hypergeometric function p Fq [z] defined by the Mellin–Barnes integral of the form

46


(1.6). The conditions will be different for the infinite contour L which has one of the following forms: (i) L = L−∞ is a left loop situated in a horizontal strip starting at the point −∞ + iϕ1 and terminating at the point −∞ + iϕ2 with −∞ < ϕ1 < ϕ2 < ∞; (ii) L = L+∞ is a right loop situated in a horizontal strip starting at the point +∞ + iϕ1 and terminating at the point +∞ + iϕ2 with −∞ < ϕ1 < ϕ2 < ∞; (iii) L = Liγ∞ is a contour starting at the point γ − i∞ and terminating at the point γ + i∞, where γ ∈ R. We suppose that ai , bj ∈ C (i = 1, · · · , p; j = 1, · · · , q) be such that bj 6= − l (j = = 1, · · · , q; l ∈ N0 ), and the poles cl = −l

(l ∈ N0 )

(2.1)

of the gamma function Γ(s) and the poles aik = ai + k

(i = 1, · · · , p; k ∈ N0 )

(2.2)

of the gamma functions Γ(ai − s) (i = 1, · · · , p) do not coincide: ai + k6= − l

(i = 1, · · · , p; k, l ∈ N0 ).

(2.3)

We also suppose that the poles aik in (2.2) are simple: ai + k6=aj + m (i6=j; i, j = 1, · · · , p; k, m ∈ N0 ).

(2.4)

We shall use the notation µ=

q X

bj −

j=1

p X

ai +

i=1

p−q . 2

(2.5)

When L = L−∞ , the existence of the extended generalized hypergeometric function p Fq [z] defined by the Mellin–Barnes integral (1.6) is given by the following result. Theorem 2.1. Let ai , bj ∈ C (i = 1, · · · , p; j = 1, · · · , q) be such that bj 6= − l (j = = 1, · · · , q; l ∈ N0 ) and the conditions in (2.3) and (2.4) are satisfied. Let one of the following conditions hold: q + 1 > p, q + 1 = p, q + 1 = p,

|z| = 1,

z6=0;

(2.6)

0 < |z| < 1;   q p X X Re  bj − ai  > 0. j=1

(2.7) (2.8)

i=1

Then there exists p Fq [z] defined by the Mellin–Barnes integral (1.6), where the path of integration L = L−∞ separates all poles cl in (2.1) to the left and all poles aik in (2.2) to the right.


47

Proof. Let Γ(s) G1,p p,q+1 (s)

=

p Y

Γ(ai − s)

i=1 q Y

.

(2.9)

Γ(bj − s)

j=1

In accordance with the Stirling formula for the gamma function [1, 1.18(2)] · µ ¶¸ 1 Γ(z) = (2π)1/2 z z−1/2 e−z 1 + O (z → ∞), (2.10) z there holds the asymptotic relation at infinity [2, (1.2.1)]: |Γ(x + iy)| ∼ (2π)1/2 |x|x−1/2 e−x−π[1−sign(x)]y/2

(x, y ∈ R; |x| → ∞). (2.11)

By applying this formula, direct calculation leads to the following asymptotic relation for the function in (2.9): µ ¶(q−p+1)|t| ¯ ¯ e ¯ 1,p ¯ |t|−[Re(µ)+1/2] ¯Gp,q+1 (t + iσ)¯ ∼ A |t|

(t, σ ∈ R, t → −∞), (2.12)

where   p q X X Re(ai ) − πσ  . A = (2π)(p−q+1)/2 exp  Re(bj ) −

(2.13)

i=1

j=1

Since (−z)−(t+iσ) = |z|−t eσ arg(−z) e−i[t arg(−z)+σ log |z|] ,

(2.14)

then, in accordance with the known convergence principle for the improper integrals, the integral in (1.6) is absolutely convergent, provided that either of conditions in (2.6), (2.7) and (2.8) is satisfied. This completes the proof of the theorem. Corollary 2.1.1. Let a1 , a2 , b1 ∈ C be such that b1 6= −l (l ∈ N0 ) and the conditions in (2.3) and (2.4) with p = 2 are satisfied. Let either of the conditions hold 0 < |z| < 1 |z| = 1,

or

Re(b1 − a1 − a2 ) > 0.

(2.15) (2.16)

Then there exists 2 F1 [a1 , a2 ; b1 ; z] defined by the Mellin–Barnes integral (1.7), where the path of integration L = L−∞ separates all poles cl in (2.1) to the left and all poles aik (i = 1, 2) in (2.2) to the right. Corollary 2.1.2. Let a1 , a2 , a3 , b1 , b2 ∈ C be such that bj 6= − l (j = 1, 2; l ∈ N0 ) and the conditions in (2.3) and (2.4) with p = 3 are satisfied. Let hold either of the

48


conditions (2.15) or |z| = 1,

Re(b1 + b2 − a1 − a2 − a3 ) > 0.

(2.17)

Then there exists 3 F2 [a1 , a2 , a3 ; b1 , b2 ; z] defined by the Mellin–Barnes integral (1.8), where the path of integration L = L−∞ separates all poles cl in (2.1) to the left and all poles aik (i = 1, 2, 3) in (2.2) to the right. The next result yields the existence of the extended generalized hypergeometric function p Fq [z] defined by the Mellin–Barnes integral (1.6) for L = L+∞ . Theorem 2.2. Let ai , bj ∈ C (i = 1, · · · , p; j = 1, · · · , q) be such that bj 6= − l (j = = 1, · · · , q; l ∈ N0 ) and the conditions in (2.3) and (2.4) are satisfied. Let one of the following conditions hold: q + 1 < p,

z6=0;

(2.18)

q + 1 = p, q + 1 = p,

|z| = 1,

|z| > 1;   q p X X Re  bj − ai  > 0. j=1

(2.19) (2.20)

i=1

Then there exists p Fq [z] defined by the Mellin–Barnes integral (1.6), where the path of integration L = L+∞ separates all poles cl in (2.1) to the left and all poles aik in (2.2) to the right. Proof. Using the relation (2.11), we deduce the following asymptotic relation for the function G1,p p,q+1 (s) in (2.9): µ ¶−(q−p+1)t ¯ ¯ e ¯ 1,p ¯ |t|−[Re(µ)+1/2] ¯Gp,q+1 (t + iσ)¯ ∼ B |t|

(t, σ ∈ R, t → +∞), (2.21)

where  B = (2π)(p+1−q)/2 e(p−q)σπ exp 

q X

[Re(bj ) + πIm(bj )] −

j=1

p X

 . [Re(ai ) + πIm(ai )](2.22)

i=1

According to (2.21), (2.14) and the known convergence principle for the improper integrals, the integral on the right hand side of (1.6) is absolutely convergent, provided that either of the conditions in (2.18), (2.19) and (2.20) is satisfied. Thus the theorem is proved. Corollary 2.2.1. Let a1 , a2 , b1 ∈ C be such that b1 6= −l (l ∈ N0 ) and the conditions in (2.3) and (2.4) with p = 2 are satisfied. Let either of the conditions (2.16) or |z| > 1.

(2.23)

hold. Then there exists 2 F1 [a1 , a2 ; b1 ; z] defined by the Mellin–Barnes integral (1.7), where the path of integration L = L+∞ separates all poles cl in (2.1) to the left and all poles aik (i = 1, 2) in (2.2) to the right. Corollary 2.2.2. Let a1 , a2 , a3 , b1 , b2 ∈ C be such that bj 6= − l (j = 1, 2; l ∈ N0 )

49


and the conditions in (2.3) and (2.4) with p = 3 are satisfied. Let either of the conditions (2.23) or (2.17) hold. Then there exists 3 F2 [a1 , a2 , a3 ; b1 , b2 ; z] defined by the Mellin–Barnes integral (1.8), where the path of integration L = L+∞ separates all poles cl in (2.1) to the left and all poles aik (i = 1, 2, 3) in (2.2) to the right. Finally we deduce the existence of the extended generalized hypergeometric function p Fq [z] defined by the Mellin–Barnes integral (1.6) for L = Liγ∞ . Theorem 2.3. Let ai , bj ∈ C (i = 1, · · · , p; j = 1, · · · , q) be such that bj 6= − l (j = = 1, · · · , q; l ∈ N0 ) and the conditions in (2.3) and (2.4) are satisfied. Let one of the following conditions hold: p + 1 > q,  p + 1 = q,

Re 

| arg(−z)| < q X j=1

bj −

p X

(p − q + 1)π , 2 

ai  > 2γ,

z6=0;

z: real negative.

(2.24)

(2.25)

i=1

Then there exists p Fq [z] defined by the Mellin–Barnes integral (1.6), where the path of integration L = Liγ∞ separates all poles cl in (2.1) to the left and all poles aik in (2.2) to the right. Proof. Using the asymptotic relation at infinity [2, (1.2.2)]: |Γ(x + iy)| ∼ (2π)1/2 |y|x−1/2 e−x−π|y|/2

(x, y ∈ R; |y| → ∞)

(2.26)

deduced from (2.10), we obtain the following asymptotic formula for G1,p p,q+1 (s): ¯ ¯ ¯ 1,p ¯ ¯Gp,q+1 (σ + it)¯ ∼ C|t|(q−p+1)σ−Re(µ)−1/2 e−(p−q+1)|t|π/2 (σ, t ∈ R, |t| → +∞), (2.27) where ³ ´ π C = (2π)(p−q+1)/2 e(p−q−1)σ−(p−q)/2 exp Re(µ) − sign(t)Im(µ) . 2

(2.28)

By (2.27), (2.14) and the known convergence principle for the improper integrals, the integral on the right hand side of (1.6) is absolutely convergent, provided that either of the conditions in (2.24) and (2.25) is satisfied. This completes the proof of the theorem. Corollary 2.3.1. Let a1 , a2 , b1 ∈ C be such that b1 6= −l (l ∈ N0 ) and the conditions in (2.3) and (2.4) with p = 2 are satisfied, and let z ∈ C be such that | arg(−z)| < π. Then there exists 2 F1 [a1 , a2 ; b1 ; z] defined by the Mellin–Barnes integral (1.7), where the path of integration L = Liγ∞ separates all poles cl in (2.1) to the left and all poles aik (i = 1, 2) in (2.2) to the right. Corollary 2.3.2. Let a1 , a2 , a3 , b1 , b2 ∈ C be such that bi 6= − l (i = 1, 2; l ∈ N0 ) and the conditions in (2.3) and (2.4) with p = 3 are satisfied, and let z ∈ C be such that | arg(−z)| < π. Then there exists 3 F2 [a1 , a2 , a3 ; b1 , b2 ; z] defined by the Mellin– Barnes integral (1.8), where the path of integration L = Liγ∞ separates all poles cl in (2.1) to the left and all poles aik (i = 1, 2, 3) in (2.2) to the right. Remark 2.1. Theorem 2.2 allows us to define the extended generalized hypergeometric function p Fq [z] defined by (1.6) with L = L+∞ for the range of

50


parameters p, q ∈ N0 such that p > q + 1 for all z ∈ C and p = q + 1 for |z| > 1. This representation can be considered as an extension of the classical generalized hypergeometric function defined by the series in (1.1) from the usual range of parameters and variable z: p < q + 1 for all z ∈ C and p = q + 1 for |z| < 1. Similarly, in the case p = q + 1 the relation (1.6) can be considered as the extension of (1.1) from |z| < 1 to |z| > 1. Remark 2.2. The above approach, giving the meaning for the extended generalized hypergeonmetric function p Fq [z] when p > q + 1, is based on the Mellin– Barnes integral representation (1.6). Another method is known to give such a meaning to the classical hypergeormetric function p Fq [a1 , · · · , ap ; b1 , · · · , bq ; z] for p > > q + 1. This approach is based on the introduction of the so-called MacRobert’s E-function (see [1, Section 5.2]). Remark 2.3. The representation (1.7) for the Gauss hypergeometric function F 2 1 [a1 , a2 ; b1 ; z], defined by the hypergeometric series (1.4), is well known (see [1, 2.1(15)]).

3 Extended generalized hypergeometric function as the Mejier G-function Now we apply the results in Theorems 2.1–2.3 of Section 2 to represent the extended generalized hypergeometric function p Fq [z] defined by (1.6) with ai , bj ∈ ∈ C (i = 1, · · · , p; j = 1, · · · , q) as a special case of the Meijer G-function Gm,n p,q (z) (see, for example, [1, Section 5.3]). This function, for m, n, p, q ∈ N0 such that 0 5 n 5 p, 1 5 m 5 q and for ai , bj ∈ C, is defined by means of a Mellin–Barnes type integral in the following manner " ¯ # Z ¯ a1 , · · · , ap 1 ¯ m,n m,n Gp,q (z)≡Gp,q z ¯ = Θ(s)z −s ds, (3.1) ¯ b1 , · · · , bq 2πi L

where m Y

Θ(s) =

j=1 p Y

Γ(bj + s)

Γ(ai + s)

i=n+1

n Y

Γ(1 − ai − s)

i=1 q Y

.

(3.2)

Γ(1 − bj − s)

j=m+1

Here z −s = exp[−s(log |z| + i arg z)],

z6=0,

i=

√

−1,

(3.3)

where log |z| represents the natural logarithm of |z| and arg z is not necessarily the principal value. An empty product in (3.2), if it occurs, is taken to be one. Let the poles bjl = −(bj + l) (j = 1, · · · , m; l ∈ N0 )

(3.4)


51

of the gamma functions Γ(bj + s) and the poles aik = 1 − ai + k

(i = 1, · · · , n; k ∈ N0 )

(3.5)

of the gamma functions Γ(1 − ai − s) do not coincide: bj + l6=ai − k − 1 (i = 1, · · · , n; j = 1, · · · , m; l, k ∈ N0 ).

(3.6)

L in (3.1) is one of the above contours L = L−∞ , L = L+∞ or L = Liγ∞ which separates all poles bjl in (3.4) to the left and all poles aik in (3.5) to the right of L. The theory of the Meijer G-function may be found in [1, Sections 5.3–5.6], [3] and [6, §8.2]. According to (1.6) and (3.1)–(3.2), we obtain the representation of the extended generalized hypergeometric function p Fq [z] as a G-function of the form q Y p Fq [a1 , · · ·

, ap ; b1 , · · · , bq ; z] =

j=1 p Y

Γ(bj )

" G1,p p,q+1

Γ(ai )

¯ # ¯ 1 − a1 , · · · , 1 − ap ¯ −z ¯ . (3.7) ¯ 0, 1 − b1 , · · · , 1 − bq

i=1

From Theorems 2.1–2.3 we deduce the conditions for these representations. Theorem 3.1. Let p, q ∈ N0 , let ai , bj ∈ C (i = 1, · · · , p; j = 1, · · · , q) be such that bj 6= − l (j = 1, · · · , q; l ∈ N0 ) and the conditions in (2.3) and (2.4) are satisfied, and let γ ∈ R. Let L be the contour which separates all poles cl in (2.1) to the left and all poles aik in (2.2) to the right. Let one of the following conditions be valid: (a) L = L−∞ and either of the conditions in (2.6), (2.7) or (2.8) holds. (b) L = L+∞ and either of the conditions in (2.18), (2.19) or (2.20) holds. (c) L = Liγ∞ and either of the conditions in (2.24) or (2.25) holds. Then the extended generalized hypergeometric function p Fq [z] defined by (1.6) is represented as a G-function by (3.7). Corollary 3.1.1. Let a1 , a2 , b1 ∈ C be such that b1 6= −l (l ∈ N0 ) and the conditions in (2.3) and (2.4) with p = 2 are satisfied, and let γ ∈ R. Let L be the contour which separates all poles cl in (2.1) to the left and all poles aik (i = 1, 2) in (2.2) to the right. Let one of the following conditions be valid: (a) L = L−∞ and either of the conditions in (2.15) or (2.16) hold. (b) L = L+∞ and either of the conditions in (2.23) or (2.16) hold. (c) L = Liγ∞ and z ∈ C is such that | arg(−z)| < π. Then 2 F1 [a1 , a2 ; b1 ; z] defined by (1.7) is represented as a G-function by " ¯ # ¯ 1 − a1 , 1 − a2 Γ(b1 ) ¯ 1,2 G −z ¯ . (3.8) 2 F1 [a1 , a2 ; b1 ; z] = ¯ 0, 1 − b1 Γ(a1 )Γ(a2 ) 2,2 Corollary 3.1.2. Let a1 , a2 , a3 , b1 , b2 ∈ C be such that bj 6= − l (j = 1, 2; l ∈ N0 ) and the conditions in (2.3) and (2.4) with p = 3 are satisfied, and let γ ∈ R. Let L be the contour which separates all poles cl in (2.1) to the left and all poles aik (i = 1, 2, 3) in (2.2) to the right. Let one of the following conditions be valid:

52


(a) L = L−∞ and either of the conditions in (2.15) or (2.17) hold. (b) L = L+∞ and either of the conditions in (2.23) or (2.17) hold. (c) L = Liγ∞ and z ∈ C is such that | arg(−z)| < π. Then 3 F2 [a1 , a2 , a3 ; b1 , b2 ; z] defined by (1.8) is represented as a G-function by " ¯ # ¯ 1 − a1 , 1 − a2 , 1 − a3 Γ(b1 )Γ(b2 ) ¯ 1,3 G −z ¯ . (3.9) 3 F2 [a1 , a2 , a3 ; b1 , b2 ; z] = ¯ 0, 1 − b1 , 1 − b2 Γ(a1 )Γ(a2 )Γ(a3 ) 3,3 Remark 3.1. The conditions (2.6) and (2.7) for the representation of p Fq [z] given in Theorem 3.1(a) coincide, except for the point z = 0, with the well known conditions of convergence of the generalized hypergeometric series (1.1) (see [1, Section 4.1]). Remark 3.2. Theorems 2.2 and 3.1(b) allow us to define the extended generalized hypergeometric function p Fq [z] by (1.6) and (3.7) for the parameters p, q and variable z ∈ C: p > q + 1 for all complex z6=0 and p = q + 1 for |z| > 1. These representations can be considered as an extension of the generalized hypergeometric function defined by the series in (1.1) from the usual range of parameters p, q and variable z: p < q + 1 for all z ∈ C and p = q + 1 for |z| < 1. Similarly, in the case p = q + 1, the relations (1.6) and (3.7) can be considered as an extension of the generalized hypergeometric function (1.1) from |z| < 1 to |z| > 1. Remark 3.3. The representation (3.8) coincides with the known formula for the Gauss hypergeometric function 2 F1 [a1 , a2 ; b1 ; z] (see, for example, [2, (2.9.15)]).

4 Series representations of extended generalized hypergeometric functions

In this section we prove the series representation for p Fq [z] defined by (1.6) and (3.7). These representations are different for L−∞ and L+∞ . From Theorem 1 we deduce the first result, which yields the series representation of the extended generalized hypergeometric function at zero. Theorem 4.1. Let p, q ∈ N0 with p 5 q + 1 and let ai , bj ∈ C (i = 1, · · · , p; j = = 1, · · · , q) be such that bj 6= − l (j = 1, · · · , q; l ∈ N0 ) and the conditions in (2.3) and (2.4) are satisfied. Let either of the condition in (2.6), (2.7) or (2.8) holds. Let L−∞ be the contour which separates all poles cl in (2.1) to the left and all poles aik in (2.2) to the right. Then p Fq [z] defined by (1.6) with L = L−∞ has the power series representation (1.1): p Y

(ai )k ∞ X zk i=1 , p Fq [a1 , · · · , ap ; b1 , · · · , bq ; z] = q Y k! k=0 (bj )k j=1

(4.1)

53


Proof. Using (2.9), we write p Fq [z] defined by (1.6) in the form q Y p Fq [a1 , · · · , ap ; b1 , · · · , bq ; z] =

j=1 p Y

Γ(bj ) Γ(ai )

1 2πi

Z −s G1,p ds. p,q+1 (s)(−z)

(4.2)

L−∞

i=1

Applying the usual procedure (see, for example, [1, Section 2.3]), we evaluate the above integral as a sum of the residues of the integrand in (4.2) at the simple poles cl in (2.1). Taking into account (2.9) and the asymptotic formula [1, 1.1(8)] Γ(z) =

(−1)k [1 + O(z + k)] (z → −k, k ∈ N0 ), k!(z + k)

(4.3)

we have q Y p Fq [a1 , · · ·

, ap ; b 1 , · · ·

j=1 , bq ; z]= p Y

 Γ(bj )

j=1 = p Y



 Γ(s)  Γ(ai − s)    i=1 −s  ress=−k  (−z)  q  Y    Γ(ai ) k=0 Γ(bj − s) ∞ X

i=1 q Y

p Y

j=1

Γ(bj )

p Y ∞ X i=1

Γ(ai ) k=0

i=1

q Y

Γ(ai + k) Γ(bj + k)

zk . k!

j=1

From here, in accordance with the formula Γ(α + k) = Γ(α)(α)k

(α ∈ C, k ∈ N0 ),

(4.4)

we find the result in (4.1). Corollary 4.1.1. Let p, q ∈ N0 with p 5 q + 1 and let ai , bj ∈ C (i = 1, · · · , p; j = = 1, · · · , q) be such that bj 6= − l (j = 1, · · · , q; l ∈ N0 ) and the conditions in (2.3) and (2.4) are satisfied. Let L−∞ be the contour which separates all poles cl in (2.1) to the left and all poles aik in (2.2) to the right. Then p Fq [z] defined by (1.6) with L = L−∞ has the following asymptotic estimate, as z → 0 : p Y

(ai )k N X zk i=1 + O(z N +1 ), p Fq [a1 , · · · , ap ; b1 , · · · , bq ; z] = q Y k! k=0 (bj )k j=1

for any N ∈ N0 .

(4.5)

54


Theorem 4.2. Let q ∈ N0 and let ai , bj ∈ C (i = 1, · · · , q + 1; j = 1, · · · , q) be such that bj 6= − l (j = 1, · · · , q; l ∈ N0 ) and the conditions in (2.3) and (2.4) are satisfied with p = q + 1. Let either of the following conditions hold: |z| < 1;

or

|z| = 1,

q X

Re(bj ) >

j=1

q+1 X

Re(ai ).

(4.6)

i=1

Let L−∞ be the contour which separates all poles cl in (2.1) to the left and all poles aik in (2.2) with p = q +1 to the right. Then the extended generalized hypergeometric function q+1 Fq [z] defined by (1.6) with L = L−∞ and p = q + 1 has the following series representation: q+1 Y q+1 Fq [a1 , · · ·

, ap ; b1 , · · · , bq ; z] =

∞ X i=1 k=0

q Y

(ai )k

(bj )k

zk , k!

(4.7)

j=1

To obtain a series representation of the extended generalized hypergeometric function (1.6) at infinity, we need a preliminary assertion which is directly verified by using the asymptotic formula (2.11).

Lemma 4.1. Let c ∈ C and k ∈ N. Then there hold the following asymptotic formula, as k → +∞, µ ¶k k |Γ(c + k)| ∼ C1 k Re(c)−1/2 , e

C1 = (2π)1/2

(4.8)

and |Γ(c − k)| ∼ C2

µ ¶−k k k Re(c)−1/2 , e

C2 = (2π)1/2 e−πIm(c) .

(4.9)

Using this lemma on the basis of Theorem 2.2 we give the series representation of the extended generalized hypergeometric function p Fq [z] in powers of −1/z.

Theorem 4.3. Let p ∈ N, q ∈ N0 with p = q + 1 and let ai , bj ∈ C (i = 1, · · · , p; j = = 1, · · · , q) be such that bj 6= − l (i = 1, · · · , q; l ∈ N0 ) and the conditions in (2.3) and (2.4) are satisfied. Let either of the condition in (2.18), (2.19) or (2.20) holds. Let L+∞ be the contour which separates all poles cl in (2.1) to the left and all poles aik in (2.2) to the right. Then p Fq [z] defined by (1.6) with L = L+∞ has the following

55


series representation: p Fq [a1 , · · · q Y

=

j=1 p Y

, ap ; b1 , · · · , bq ; z]

Γ(bj ) Γ(ai )

p X ∞ X

=

j=1 p Y

Γ(ai − ah − k)

i=1,i6=h q Y

h=1 k=0

i=1 q Y

p Y

Γ(ah + k)

Γ(bj − ah − k)

(−1)k k!

µ ¶a +k 1 h − z

(4.10)

j=1

Γ(bj ) Γ(ai )

p X

Γ(ah )

h=1

i=1

p Y

i=1,i6=h q Y

Γ(ai − ah ) µ

1 − z

Γ(bj − ah )

Ã

¶ ah q+1 Fp−1

ah , 1 + ah − b1 ,

j=1

! (−1)p−q+1 · · · , 1 + ah − bq ; 1 + ah − a1 , ·˘· ·, 1 + ah − ap ; , z ∗

(4.11)

where the asterisk in (4.11) indicates the omission of the parameter 1 + ah − ah (h = 1, · · · , p).

Proof. By Theorem 2.2, the relation (1.6) is valid where the contour L = L+∞ separates all poles cl in (2.1) to the left and all poles ahk = ah + k (h = 1, · · · , p; k ∈ ∈ N0 ) to the right. We evaluate (1.6) as a sum of residues of the integrand in (1.6) at the points ahk : q Y p Fq [a1 , · · ·

, ap ; b1 , · · ·

j=1 , bq ; z]= p Y

Γ(bj ) Γ(ai )

1 2πi

Z

j=1 =− p Y

Γ(ai − s)

i=1 q Y L+∞

i=1 q Y

Γ(s)

p Y

(−z)s ds

(4.12)

Γ(bj − s)

j=1

 Γ(bj )

p Y



  Γ(s) Γ(ai − s)     i=1 (−z)s  . ress=ahk  q   Y   Γ(ai ) h=1 k=0 Γ(bj − s)

i=1

p X ∞ X

j=1

By (2.4), all poles ahk = ah +k are simple. Calculating the above residues and taking into account the asymptotic formula Γ(ah − s) ∼

(−1)k+1 k!(s − ah − k)

(s → ah + k; i = 1, · · · , p, k ∈ N0 )

(4.13)

which follows from the asymptotic formula (4.3), we obtain the formula (4.10).

56


Now we prove the convergence of the series in (4.10). Let us consider the series µ ¶a ∞ ¶k µ 1 hX 1 − , chk − z z

(4.14)

k=0

where p Y

chk =

Γ(ai − ah − k)

i=1,i6=h q Y

Γ(ah + k)

Γ(bj − ah − k)

(−1)k k!

(j = 1, · · · , p).

(4.15)

j=1

Using the relations (4.8) and (4.9), we obtain the asymptotic relation for chk , as k → +∞ |chk | ∼ Ch

µ ¶(q−p+1)k k k (1+q−p)Re(ah )−Re(µ)−1/2 , e

(4.16)

where Ch = (2π)(p−q−1)/2 exp ([Im(µ + ah (p − q))] π)

(4.17)

and µ is given in (2.3). Now the convergence of the series in (4.14) follows from the known convergence principle for the power series, which completes the proof of the relation (4.10). The relation (4.11) is derived from (4.10) by using the directly verified formula Γ(a − k) = (−1)k

Γ(a) (a ∈ C, k ∈ N0 ). (1 − a)k

(4.18)

This completes the proof of the theorem. Corollary 4.3.1. Let p ∈ N, q ∈ N0 with p = q +1 and let ai , bj ∈ C (i = 1, · · · , p; j = = 1, · · · , q) be such that bj 6= − l (j = 1, · · · , q; l ∈ N0 ) and the conditions in (2.3) and (2.4) are satisfied. Let L+∞ be the contour which separates all poles cl in (2.1) to the left and all poles aik in (2.2) to the right. Then p Fq [z] defined by (1.6) with L = L+∞ has the following asymptotic estimate, as z → ∞ : p Fq [a1 , · · · q Y

=

j=1 p Y

, ap ; b1 , · · · , bq ; z]

Γ(bj )

(

p N X X

Γ(ai ) h=1

i=1

k=0

p Y

Γ(ah + k)

Γ(ai − ah − k)

i=1,i6=h q Y

Γ(bj − ah − k)

(−1)k k!

µ −

1 z

¶ah +k

j=1

³

+O z −Re(ah )−N −1

´

) ,

(4.19)

57


for any N ∈ N0 . In particular, as z → ∞, p Fq [a1 , · · ·

, ap ; b1 , · · · , bq ; z]   p q Y Y      Γ(ah )  Γ(ai − ah ) µ Γ(bj )     ¶ p ah   ³ ´ X 1 j=1 i=1,i6=h −Re(ah )−1 = p − +O z (4.20) q Y Y   z  h=1      Γ(ai ) Γ(bj − ah )     i=1

j=1

and p Fq [a1 , · · ·

¡ ¢ , ap ; b1 , · · · , bq ; z] = O z −ω ,

(4.21)

where ω = min [Re(ah )].

(4.22)

15h5p

When p = q + 1, Theorem 4.2 yields the following result. Theorem 4.4. Let q ∈ N0 and let ai , bj ∈ C (i = 1, · · · , q + 1; j = 1, · · · , q) be such that bj 6= − l (j = 1, · · · , q; l ∈ N0 ) and the conditions in (2.3) and (2.4) are satisfied with p = q + 1. Let either of the following conditions hold: |z| > 1;

or

|z| = 1,

q X

Re(bj ) >

q+1 X

Re(ai ).

(4.23)

i=1

j=1

Let L+∞ be the contour which separates all poles cl in (2.1) to the left and all poles aik in (2.2) with p = q +1 to the right. Then the extended generalized hypergeometric function q+1 Fq [z] defined by (1.6) with L = L+∞ and p = q + 1 has the following series representation: q+1 Fq [a1 , · · · q Y

=

, aq+1 ; b1 , · · · , bq ; z] Γ(bj )

j=1 q+1 Y

Γ(ai )

q+1 X ∞ X

=

q Y

h=1 k=0

i=1

·

Γ(bj − ah − k)

(−1)k k!

µ ¶a +k 1 h − (4.24) z

j=1

Γ(bj )

j=1 q+1 Y

Γ(ai − ah − k)

i=1,i6=h

i=1 q Y

q+1 Y

Γ(ah + k)

Γ(ai )

q+1 X h=1

Γ(ah )

q+1 Y

i=1,i6=h q Y

Γ(ai − ah ) µ

Γ(bj − ah )

1 − z

¶ah

j=1

µ ¶ ∗ 1 ˘ F a , 1 + a − b , · · · , 1 + a − b ; 1 + a − a , · · ·, 1 + a − a ; , (4.25) q+1 q h h 1 h q h 1 h q+1 z

58


where the asterisk in (4.25) indicates the omission of the parameter 1 + ah − ah (h = 1, · · · , q + 1). Corollary 4.4.1. Let q ∈ N0 and let ai , bj ∈ C (i = 1, · · · , q +1; j = 1, · · · , q) be such that bj 6= − l (j = 1, · · · , q; l ∈ N0 ) and the conditions in (2.3) and (2.4) are satisfied with p = q + 1. Let L+∞ be the contour which separates all poles cl in (2.1) to the left and all poles aik in (2.2) with p = q + 1 to the right. Then q+1 Fq [z] defined by (1.6) with L = L+∞ and p = q + 1 has the asymptotic estimate, as z → ∞, q+1 Fq [a1 , · · · q Y

, aq+1 ; b1 , · · · , bq ; z] q+1 Y

Γ(ai − ah − k) ( N Γ(ah + k) µ ¶a +k q+1 X X (−1)k 1 h i=1,i6=h = q+1 − q Y k! z Y h=1 k=0 Γ(bj − ah − k) Γ(ai ) Γ(bj )

j=1

j=1

i=1

³

+O z −Re(ah )−N −1

´

) (4.26)

for any N ∈ N0 . In particular, q+1 Fq [a1 , · · ·

, aq+1 ; b1 , · · · , bq ; z]   q+1 q Y Y       Γ(ai − ah ) µ Γ(bj )     ¶ p ah X  ³ ´ 1 j=1 i=1,i6=h −Re(ah )−1 = q+1 Γ(ah ) q − +O z (4.27) Y   z Y   h=1     Γ(bj − ah ) Γ(ai )     j=1

i=1

and q+1 Fq [a1 , · · ·

¡ ¢ , aq+1 ; b1 , · · · , bq ; z] = O z −ω

,

(4.28)

where ω=

min [Re(ah )].

15h5q+1

(4.29)

Remark 4.1. The formula of the form (4.25) is known for the classical hypergeometric function q+1 Fq [a1 , · · · , ap ; b1 , · · · , bq ; z]; see [6, 7.2.3.77].

5 Asymptotic expansions of extended generalized hypergeometric functions In Section 4 we have established the power series representations (4.1) and (4.10) for the extended generalized hypergeometric function p Fq [z] defined by the Mellin– Barnes integral (1.6) and have deduced the asymptotic estimates (4.5) and (4.19) near zero and infinity. Really, the relation (4.1) yields the asymptotic expansion of

59

A. KILBAS, R. SAXENA, M. SAIGO, J. TRUJILLO p Fq [z]

near zero: p Y

(ai )k ∞ X zk i=1 , p Fq [a1 , · · · , ap ; b1 , · · · , bq ; z] ∼ q Y k! k=0 (bj )k

(5.1)

j=1

as z → 0, provided that L = L−∞ and p 5 q + 1. Similarly (4.10) and (4.11) yield the asymptotic expansions of p Fq [z] near infinity: p Fq [a1 , · · · q Y

∼

j=1 p Y

, ap ; b1 , · · · , bq ; z]

Γ(bj ) Γ(ai )

p X ∞ X

=

j=1 p Y

Γ(ai − ah − k)

i=1,i6=h q Y

h=1 k=0

i=1 q Y

p Y

Γ(ah + k)

Γ(bj − ah − k)

(−1)k k!

µ ¶a +k 1 h − z

(5.2)

j=1

Γ(bj ) Γ(ai )

p X

Γ(ah )

h=1

i=1

p Y

i=1,i6=h q Y

Γ(ai − ah ) µ

1 − z

Γ(bj − ah )

Ã

¶ ah q+1 Fp−1

ah , 1 + ah − b1 ,

j=1

! (−1)p−q+1 · · · , 1 + ah − bq ; 1 + ah − a1 , ·˘· ·, 1 + ah − ap ; , z ∗

(5.3)

as z → ∞, provided that L = L+∞ and p = q + 1. The asterisk in (5.3) indicates the omission of the parameter 1 + ah − ah (h = 1, · · · , p). These formulas remain true for a wide range of parameters p and q. For this m,n reason we can apply the known asymptotic results for the H-function Hp,q (z) presented in [2, Sections 1.5 and 1.8]. Such a function is defined for ai , bj ∈ C and αi , βj > 0 (i = 1, · · · , p, j = 1, · · · , q) by the Mellin–Barnes integral " ¯ # Z ¯ (a1 , α1 ), · · · , (ap , αp ) 1 ¯ m,n m,n Hp,q (z)≡Hp,q z ¯ Θ(s)z −s ds, (5.4) = ¯ (b1 , β1 ), · · · , (bq , βq ) 2πi L

where m Y

Θ(s) =

j=1 p Y

Γ(bj + βj s)

Γ(ai + αi s)

i=n+1

n Y

Γ(1 − ai − αi s)

i=1 q Y j=m+1

. Γ(1 − bj − βj s)

(5.5)

60


An empty product in (5.5), if it occurs, is taken to be one, and L is one of the contours L−∞ , L+∞ and Liγ∞ defined in Section 2, which separates all poles of the gamma functions Γ(bj + βj s) (j = 1, · · · , m) to the left and all poles of gamma functions Γ(1 − ai − αi s) (i = 1, · · · , n) to the right. A detailed and comprehensive account of the H-function is available from the monographs by Mathai and Saxena [5], by Srivastava, Gupta and Goyal [10], by Prudnikov, Brychkov and Marichev [6, §8.3] and by Kilbas and Saigo [2, Chapters 1–2]. The Meijer G-function (3.1) is a special case of the H-function: " ¯ # " ¯ # ¯ a1 , · · · , ap ¯ (a1 , 1), · · · , (ap , 1) ¯ ¯ m,n m,n Gp,q z ¯ ≡ Hp,q z ¯ . ¯ b1 , · · · , bq ¯ (b1 , 1), · · · , (bq , 1)

(5.6)

By (3.7) and (5.6), the extended generalized hypergeometric function p Fq [z] is the function G1,p p,q+1 (−z) of the form q Y p Fq [a1 , · · ·

, ap ; b1 , · · · , bq ; z] =

j=1 p Y

Γ(bj )

" G1,p p,q+1

Γ(ai )

¯ # ¯ 1 − a1 , · · · , 1 − ap ¯ −z ¯ . (5.7) ¯ 0, 1 − b1 , · · · , 1 − bq

i=1

The asymptotic behaviors of the H-function (5.4) at infinity and zero depend on parameters ∆ and a∗ defined by ∆=

q X j=1

βj −

p X

αi , a∗ =

i=1

n X

αi −

i=1

p X i=n+1

αi +

m X j=1

βj −

q X

βj .

(5.8)

j=m+1

m,n Lemma 5.1. [2, Theorem 1.7] If either ∆ 5 0 or ∆ > 0, a∗ > 0, then Hp,q (z) has the following asymptotic expansion near infinity: m,n Hp,q (z) ∼

n X ∞ X (−1)k i=1 k=0

k!αi

hik z (ai −1−k)/αi (z → ∞)

(5.9)

with the additional condition | arg(z)| < a∗ π/2 in the case ∆ > 0, a∗ > 0, where ¶ Y ¶ µ µ m n Y βj αj Γ bj + [1 − ai + k] Γ 1 − aj − [1 − ai + k] αi αi j=1 j=1,j6=i hik = p µ ¶ ¶ . (5.10) µ q Y Y αj βj Γ aj + [1 − ai + k] Γ 1 − bj − [1 − ai + k] αi j=m+1 αi j=n+1 1,p ∗ For the function G1,p p,q+1 (−z) = Hp,q+1 (−z) in (5.7), the constants ∆ and a in (5.8) are given by

∆ = q + 1 − p, a∗ = p + 1 − q,

(5.11)

61


and the condition | arg(z)| < a∗ π/2 takes the form | arg(−z)| < (p + 1 − q)π/2. It follows from (5.6) that the expansion (5.9) for G1,p p,q+1 (−z) takes the form " G1,p p,q+1

∼

¯ # ¯ 1 − a1 , · · · , 1 − ap ¯ −z ¯ ¯ 0, 1 − b1 , · · · , 1 − bq

p X ∞ X

p Y

Γ(ah + k)

Γ(ai − ah − k)

i=1,i6=h q Y

h=1 k=0

Γ(bj − ah − k)

(−1)k k!

µ ¶a +k 1 h − , z

(5.12)

j=1

provided that either q + 1 − p 5 0 or q + 1 − p > 0, p + 1 − q > 0. This means q 5 p − − 1 and p − 1 < q < p + 1, i.e. q = p, respectively and hence q 5 p. Substituting the expansion (5.12) into (5.7) we deduce the asymptotic expansion (5.2) for the generalized hypergeometric function p Fq [z] at infinity. By (4.18), (5.2) is equivalent to (5.3). Thus we obtain the following result. Theorem 5.1. Let p ∈ N0 , q ∈ N0 with p = q and let ai , bj ∈ C (i = 1, · · · , p; j = = 1, · · · , q) be such that bj 6= − l (i = 1, · · · , q; l ∈ N0 ) and the conditions in (2.3) and (2.4) are satisfied. Then the extended generalized hypergeometric function p Fq [z] has the asymptotic expansions (5.2) and (5.3), as z → ∞, where | arg(−z)| < π/2 when p = q. In particular, there hold the asymptotic estimates (4.19), (4.20) and (4.21) with ω being given by (4.22). m,n (z) Lemma 5.2. [2, Theorem 1.11] If either ∆ = 0 or ∆ < 0, a∗ > 0, then Hp,q has the following asymptotic expansion near zero: m,n Hp,q (z) ∼

m X ∞ X

h∗jl z (bj +l)/βj (z → 0)

(5.13)

j=1 l=0

with the additional condition | arg(z)| < a∗ π/2 in the case ∆ < 0, a∗ > 0, where ¶ n µ ¶ αi βi Y Γ 1 − ai + [bj + l] Γ bi − [bj + l] βj i=1 βj (−1)l i=1,i6=j ∗ hjl = ¶ Y ¶. µ µ q p l!βj Y βi αi Γ 1 − bi + [bj + l] Γ ai − [bj + l] βj i=m+1 βj i=n+1 m Y

µ

(5.14)

It follows from (5.6) that the expansion (5.13) for G1,p p,q+1 (−z) takes the form " G1,p p,q+1

p Y ¯ # Γ(ai + k) ∞ ¯ 1 − a1 , · · · , 1 − ap X zk ¯ i=1 , −z ¯ ∼ q Y ¯ 0, 1 − b1 , · · · , 1 − bq k! k=0 Γ(bj + k)

(5.15)

j=1

provided that either q + 1 − p = 0 or q + 1 − p < 0, p + 1 − q > 0 due to (5.11). It

62


means that p 5 q + 1 or p = q + 2, respectively, and with no constraint between p and q. Substituting the expansion (5.15) into (5.7) and taking (4.4) into account, we deduce the asymptotic expansion (5.1) for p Fq [z] at zero. Thus the following result holds: Theorem 5.2. Let p, q ∈ N0 , and let ai , bj ∈ C (j = 1, · · · , p; j = 1, · · · , q) be such that bj 6= − l (i = 1, · · · , q; l ∈ N0 ) and the conditions in (2.3) and (2.4) are satisfied. Then the extended generalized hypergeometric function p Fq [z] has the asymptotic expansion (5.1), as z → 0. In particular, there holds the asymptotic estimate (4.5). of

When p = q + 1, from Theorems 5.1 and 5.2 we obtain the asymptotic expansion at both infinity and zero.

q+1 Fq [z]

Theorem 5.3. Let q ∈ N0 , and let ai , bj ∈ C (i = 1, · · · , q + 1; j = 1, · · · , q) be such that bj 6= − l (i = 1, · · · , q; l ∈ N0 ) and the conditions in (2.3) and (2.4) with p = q + 1 are satisfied. (i) The extended generalized hypergeometric function asymptotic expansions, as z → ∞, q+1 Fq [a1 , · · · q Y

∼

Γ(bj ) Γ(ai )

q+1 X ∞ X

=

i=1

Γ(ai − ah − k)

i=1,i6=h q Y

h=1 k=0

Γ(bj − ah − k)

µ ¶a +k 1 h − z

(−1)k k!

(5.16)

j=1

Γ(bj )

j=1 q+1 Y

q+1 Y

Γ(ah + k)

i=1 q Y

has the

, aq+1 ; b1 , · · · , bq ; z]

j=1 q+1 Y

q+1 Fq [z]

Γ(ai )

q+1 X h=1

Γ(ah )

q+1 Y

i=1,i6=h q Y

Γ(ai − ah ) µ

1 − z

Γ(bj − ah )

Ã

¶ah q+1 Fq

ah , 1 + ah − b1 ,

j=1

! 1 · · · , 1 + ah − bq ; 1 + ah − a1 , ·˘· ·, 1 + ah − aq+1 ; . z ∗

(5.17)

The asterisk in (5.17) indicates the omission of the parameter 1 + ah − ah (h = = 1, · · · , q + 1). In particular, there hold the asymptotic estimates (4.26), (4.27) and (4.28) with ω being given by (4.29). (ii) The extended generalized hypergeometric function asymptotic expansion, as z → 0,

q+1 Fq [z]

has the

q+1 Y q+1 Fq [a1 , · · ·

, aq+1 ; b1 , · · · , bq ; z] ∼

∞ X i=1 k=0

q Y j=1

(ai )k

(bj )k

zk . k!

(5.18)


63

In particular, for any N ∈ N0 , as z → 0, q+1 Y q+1 Fq [a1 , · · ·

, aq+1 ; b1 , · · · , bq ; z] =

(ai )k

N X i=1 k=0

q Y

(bj )k

zk + O(z N +1 ). k!

(5.19)

j=1

Remark 5.1. When some poles of the gamma functions Γ(ai −s) (i = 1, · · · , p) are not simple (i.e. the conditions in (2.3) are not satisfied), then the asymptotic series expansions (5.1) and (5.2) are replaced by asymptotic power-logarithmic expansions for the function G1,p p,p+1 (−z) in (5.7). The explicit formulas can be deduced from the m,n known asymototic power-logarithmic expansions for the function Hp,q (z); see [2, Theorems 1.12 and 1.8]. Remark 5.2. The series representations (4.1) and (4.7) can be also deduced from the corresponding series representations for the function G1,p p,q+1 (−z) in (5.7) by using the known series expansions for the function Gm,n p,q (z); for example, see [2, Theorems 1.3 and 1.4]. When some poles of the gamma functions Γ(ai − s) (i = = 1, · · · , p) are not simple (i.e. the conditions in (2.3) are not satisfied), then the series representations (4.1) and (4.10) are replaced by asymptotic power-logarithmic expansions for the function G1,p p,p+1 (−z) in (5.7). The explicit formulas can be m,n deduced from the known power-logarithmic expansions for the function Hp,q (z); see [2, Theorems 1.5 and 1.6].

6 Special cases In this section we present special cases of the series representations and asymptotic expansions for the extended generalized hypergeometric functions established in Sections 4 and 5. We begin from the case p = 2 and q = 1, when Theorem 4.3 and 5.3 yield the corresponding results for the extended hypergeometric function 2 F1 [a1 , a2 ; b1 ; z]. Theorem 6.1. Let a1 , a2 , b1 ∈ C be such that b1 6= − l,

a1 + k6= − l,

a2 + m6= − l,

a1 + k6=a2 + m (k, l, m ∈ N0 ).

(6.1)

(i) Let either of the conditions in (2.15) or (2.16) hold, and L = L−∞ be the contour which separates all poles cl in (2.1) to the left and all poles a1 + k and a2 + m (k, m ∈ N0 ) to the right. Then 2 F1 [a1 , a2 ; b1 ; z] defined by the Mellin–Barnes integral (1.7) with L = L−∞ has the power series representation (1.4). (ii)

2 F1 [a1 , a2 ; b1 ; z]

has the asymptotic expansion, as z → 0,

2 F1 [a1 , a2 ; b1 ; z] ∼

∞ X (a1 )k (a2 )k z k . (b1 )k k!

k=0

(6.2)

64


In particular, for any N ∈ N0 , 2 F1 [a1 , a2 ; b1 ; z]

=

N X ¡ ¢ (a1 )k (a2 )k z k + O z N +1 (b1 )k k!

(z → 0).

(6.3)

k=0

Theorem 6.2. Let a1 , a2 , b1 ∈ C be such that the conditions in (6.1) are satisfied. (i) Let either of the conditions in (2.16) or (2.23) hold, and let L = L+∞ be the contour which separates all poles cl in (2.1) to the left and all poles a1 + k and a2 + m (k, m ∈ N0 ) to the right. Then for the extended hypergeometric function 2 F1 [a1 , a2 ; b1 ; z] defined by the Mellin–Barnes integral (1.7) with L = L+∞ , there holds the formula · µ ¶a ¸ Γ(b1 )Γ(a2 − a1 ) 1 1 1 − 2 F1 [a1 , a2 ; b1 ; z]= 2 F1 a1 , 1 + a1 − b1 ; 1 + a1 − a2 ; Γ(a2 )Γ(b1 − a1 ) z z · µ ¶a2 ¸ 1 Γ(b1 )Γ(a1 − a2 ) 1 + − . (6.4) 2 F1 a2 , 1 + a2 − b1 ; 1 + a2 − a1 ; Γ(a1 )Γ(b1 − a2 ) z z (ii)

2 F1 [a1 , a2 ; b1 ; z]

has the asymptotic expansion, as z → ∞,

µ ¶a ∞ µ ¶k Γ(b1 )Γ(a2 − a1 ) 1 1 X (a1 )k (1 + a1 − b1 )k 1 − 2 F1 [a1 , a2 ; b1 ; z]∼ Γ(a2 )Γ(b1 − a1 ) z k!(1 + a1 − a2 )k z k=0

µ ¶a ∞ µ ¶k Γ(b1 )Γ(a1 − a2 ) 1 2 X (a2 )k (1 + a2 − b1 )k 1 + − . Γ(a1 )Γ(b1 − a2 ) z k!(1 + a2 − a1 )k z

(6.5)

k=0

Corollary 6.2.1. Let a1 , a2 , b1 ∈ C be such that the conditions in (6.1) are satisfied. Then 2 F1 [a1 , a2 ; b1 ; z] has the following asymptotic estimate, as z → ∞, µ ¶a ³ ´ Γ(b1 )Γ(a2 − a1 ) 1 1 − + O z −Re(a1 )−1 2 F1 [a1 , a2 ; b1 ; z] = Γ(a2 )Γ(b1 − a1 ) z µ ¶a ³ ´ Γ(b1 )Γ(a1 − a2 ) 1 2 + − + O z −Re(a2 )−1 . (6.6) Γ(a1 )Γ(b1 − a2 ) z In particular, 2 F1 [a1 , a2 ; b1 ; z]

¡ ¢ = O z −ω

(z → ∞),

(6.7)

where ω = min[Re(a1 ), Re(a2 )].

(6.8)

Remark 6.1. For the Gauss hypergeometric function 2 F1 [a1 , a2 ; b1 ; z] the formula of the form (6.4) is well known as the analytic continuation of (1.4) to the domain | arg(−z)| < π, and the relation of the form (6.6) is a very useful formula for finding the asymptotic expansions of hypergeometric functions of two or more variables and generalized elliptic-type integrals. In this connection the reader is referred to

65


the papers by Saxena and Kalla [7], Saxena, Kalla and Hubbell [8] and Saxena and Pathan [9]. Remark 6.2. The asymptotic formula of the form (6.7) for 2 F1 [a1 , a2 ; b1 ; z] is well known (see, for example, [1, 2.3(9)]). When p = q = 2, then (1.6) and (5.7) take the forms Z 1 Γ(s)Γ(a1 − s)Γ(a2 − s) −s z ds 2 F2 [a1 , a2 ; b1 , b2 ; z]= 2πi Γ(b1 − s)Γ(b2 − s) L

" ¯ # ¯ 1 − a1 , 1 − a2 , Γ(b1 )Γ(b2 ) 1,2 ¯ = −z ¯ G . ¯ 0, 1 − b1 , 1 − b2 Γ(a1 )Γ(a2 ) 2,3

(6.9)

From Theorems 4.1, 5.2 and 5.1, we obtain the following results for the extended generalized hypergeometric function 2 F2 [a1 , a2 ; b1 , b2 ; z]: Theorem 6.3. Let ai , bj ∈ C (i, j = 1, 2) be such that bj 6= − l,

ai + k6= − l

(i = 1, 2; j = 1, 2; k, l ∈ N0 ),

ai + k6=aj + m (i6=j; i, j = 1, 2; k, m ∈ N0 ).

(6.10)

(i) Let z6=0, and let L = L−∞ be the contour which separates all poles cl in (2.1) to the left and all poles ai + k (i = 1, 2; k ∈ N0 ) to the right. Then 2 F2 [a1 , a2 ; b1 , b2 ; z] has the series representation 2 F2 [a1 , a2 ; b1 , b2 ; z] =

∞ X (a1 )k (a2 )k z k . (b1 )k (b2 )k k!

(6.11)

k=0

(ii)

2 F2 [a1 , a2 ; b1 , b2 ; z]


2 F2 [a1 , a2 ; b1 , b2 ; z] ∼

∞ X (a1 )k (a2 )k z k . (b1 )k (b2 )k k!

(6.12)

k=0

In particular, for any N ∈ N0 , the following asymptotic formula holds 2 F2 [a1 , a2 ; b1 , b2 ; z]

=

N X ¢ ¡ (a1 )k (a2 )k z k + O z N +1 (b1 )k (b2 )k k!

(z → 0).

(6.13)

k=0

Theorem 6.4. Let ai , bj ∈ C (i, j = 1, 2) be such that the conditions in (6.10) are satisfied. Then 2 F2 [a1 , a2 ; b1 , b2 ; z] has the asymptotic expansion, as z → ∞,

66


| arg(−z)| < π/2, 2 F2 [a1 , a2 ; b1 , b2 ; z]∼

µ ¶a Γ(b1 )Γ(b2 )Γ(a2 − a1 ) 1 1 − Γ(a2 )Γ(b1 − a1 )Γ(b2 − a1 ) z ·

µ ¶k ∞ X 1 (a1 )k (1 + a1 − b1 )k (1 + a1 − b2 )k − k!(1 + a1 − a2 )k z

k=0

µ ¶a Γ(b1 )Γ(b2 )Γ(a1 − a2 ) 1 2 + − Γ(a1 )Γ(b1 − a2 )Γ(b2 − a2 ) z µ ¶k ∞ X (a2 )k (1 + a2 − b1 )k (1 + a2 − b2 )k 1 · − . (6.14) k!(1 + a2 − a1 )k z k=0

Corollary 6.4.1. Let a1 , a2 , b1 , b2 ∈ C be such that the conditions in (6.10) are satisfied. Then 2 F2 [a1 , a2 ; b1 , b2 ; z] has the following asymptotic estimate, as z → ∞, | arg(−z)| < π/2, µ ¶a ³ ´ Γ(b1 )Γ(b2 )Γ(a2 − a1 ) 1 1 + O z −Re(a1 )−1 − 2 F2 [a1 , a2 ; b1 , b2 ; z]= Γ(a2 )Γ(b1 − a1 )Γ(b2 − a1 ) z µ ¶a ³ ´ Γ(b1 )Γ(b2 )Γ(a1 − a2 ) 1 2 + + O z −Re(a2 )−1 (6.15) . − Γ(a1 )Γ(b1 − a2 )Γ(b2 − a2 ) z In particular, 2 F2 [a1 , a2 ; b1 , b2 ; z]

¡ ¢ = O z −ω

(z → ∞),

(6.16)

where ω is given in (6.8). When p = 3 and q = 2, from Theorems 4.3 and 5.3 we deduce the result for the generalized hypergeometric function 3 F2 [a1 , a2 , a3 ; b1 , b2 ; z]. Theorem 6.5. Let a1 , a2 , a3 , b1 , b2 ∈ C be such that bj 6= − l,

ai + k6= − l

(i = 1, 2, 3; j = 1, 2; k, l ∈ N0 ),

ai + k6=aj + m (i6=j; i, j = 1, 2, 3, k, m ∈ N0 ).

(6.17)

(i) Let either of the conditions in (2.15) or (2.17) hold, and let L = L−∞ be the contour which separates all poles cl in (2.1) to the left and and all poles ai + + k (i = 1, 2, 3; k ∈ N0 ) to the right. Then the extended generalized hypergeometric function 3 F2 [a1 , a2 , a3 ; b1 , b2 ; z] defined by the Mellin–Barnes integral (1.8) with L = = L−∞ has the power series representation (1.5). (ii)

3 F2 [a1 , a2 , a3 ; b1 , b2 ; z]


3 F2 [a1 , a2 , a3 ; b1 , b2 ; z]

∼

∞ X (a1 )k (a2 )k (a3 )k z k . (b1 )k (b2 )k k!

k=0

In particular, for any N ∈ N0 ,

(6.18)

67


3 F2 [a1 , a2 , a3 ; b1 , b2 ; z] =

N X ¡ ¢ (a1 )k (a2 )k (a3 )k z k + O z N +1 (b1 )k (b2 )k k!

(z → 0). (6.19)

k=0

Theorem 6.6. Let a1 , a2 , a3 , b1 , b2 ∈ C be such that the conditions in (6.17) are satisfied.

(i) Let either of the conditions in (2.17) or (2.23) hold, and let L+∞ be the contour which separates all poles cl in (2.1) to the left and all poles ai + k (i = = 1, 2, 3; k ∈ N0 ) to the right. Then for the extended generalized hypergeometric function 3 F2 [a1 , a2 , a3 ; b1 , b2 ; z] defined by the Mellin–Barnes integral (1.8) with L = = L+∞ there holds the formula

3 F2 [a1 , a2 , a3 ; b1 , b2 ; z]=

µ ¶a Γ(b1 )Γ(b2 )Γ(a2 − a1 )Γ(a3 − a1 ) 1 1 − Γ(a2 )Γ(a3 )Γ(b1 − a1 )Γ(b2 − a1 ) z

¸ · 1 × 3 F2 a1 , 1 + a1 − b1 , 1 + a1 − b2 ; 1 + a1 − a2 , 1 + a1 − a3 ; z µ ¶a2 Γ(b1 )Γ(b2 )Γ(a1 − a2 )Γ(a3 − a2 ) 1 + − Γ(a1 )Γ(a3 )Γ(b1 − a2 )Γ(b2 − a2 ) z µ ¶ 1 × 3 F2 a2 , 1 + a2 − b1 , 1 + a2 − b2 ; 1 + a2 − a1 , 1 + a2 − a3 ; z µ ¶a3 Γ(b1 )Γ(b2 )Γ(a1 − a3 )Γ(a2 − a3 ) 1 + − Γ(a1 )Γ(a2 )Γ(b1 − a3 )Γ(b2 − a3 ) z ¸ · 1 × 3 F2 a3 , 1 + a3 − b1 , 1 + a3 − b2 ; 1 + a3 − a1 , 1 + a3 − a2 ; . (6.20) z

68


(ii)

3 F2 [a1 , a2 , a3 ; b1 , b2 ; z]

has the asymptotic expansion, as z → ∞, µ ¶a Γ(b1 )Γ(b2 )Γ(a2 − a1 )Γ(a3 − a1 ) 1 1 F [a , a , a ; b , b ; z]∼ − 3 2 1 2 3 1 2 Γ(a2 )Γ(a3 )Γ(b1 − a1 )Γ(b2 − a1 ) z ×

µ ¶k ∞ X (a1 )k (1 + a1 − b1 )k (1 + a1 − b2 )k 1 k=0

k!(1 + a1 − a2 )k (1 + a1 − a3 )k

z

µ ¶a Γ(b1 )Γ(b2 )Γ(a1 − a2 )Γ(a3 − a2 ) 1 2 + − Γ(a1 )Γ(a3 )Γ(b1 − a2 )Γ(b2 − a2 ) z ×

µ ¶k ∞ X (a2 )k (1 + a2 − b1 )k (1 + a2 − b2 )k 1 k=0

+

k!(1 + a2 − a1 )k (1 + a2 − a3 )k

z

µ ¶a 1 3 Γ(b1 )Γ(b2 )Γ(a1 − a3 )Γ(a2 − a3 ) − Γ(a1 )Γ(a2 )Γ(b1 − a3 )Γ(b2 − a3 ) z ×

µ ¶k ∞ X (a3 )k (1 + a3 − b1 )k (1 + a3 − b2 )k 1 k=0

k!(1 + a3 − a1 )k (1 + a3 − a2 )k

z

.

(6.21)

Corollary 6.6.1. Let a1 , a2 , a3 , b1 , b2 ∈ C be such that the conditions in (6.17) are satisfied. Then 3 F2 [a1 , a2 , a3 ; b1 , b2 ; z] has the following asymptotic estimate, as z → ∞, µ ¶a ³ ´ 1 1 Γ(b1 )Γ(b2 )Γ(a2 − a1 )Γ(a3 − a1 ) − + O z −Re(a1 )−1 F [a , a , a ; b , b ; z] = 3 2 1 2 3 1 2 Γ(a2 )Γ(a3 )Γ(b1 − a1 )Γ(b2 − a1 ) z µ ¶a2 ´ ³ Γ(b1 )Γ(b2 )Γ(a1 − a2 )Γ(a3 − a2 ) 1 + − + O z −Re(a2 )−1 Γ(a1 )Γ(a3 )Γ(b1 − a2 )Γ(b2 − a2 ) z µ ¶a ´ ³ Γ(b1 )Γ(b2 )Γ(a1 − a3 )Γ(a2 − a3 ) 1 3 + − (6.22) + O z −Re(a3 )−1 . Γ(a1 )Γ(a2 )Γ(b1 − a3 )Γ(b2 − a3 ) z In particular, 3 F2 [a1 , a2 , a3 ; b1 , b2 ; z]

¢ ¡ = O z −ω

(z → ∞),

(6.23)

where ω = min[Re(a1 ), Re(a2 ), Re(a3 )].

(6.24)

69


Finally, when p = q = 3, then (1.6) and (5.7) take the forms Z 1 Γ(s)Γ(a1 − s)Γ(a2 − s)Γ(a3 − s) −s z ds (6.25) 3 F3 [a1 , a2 , a3 ; b1 , b2 , b3 ; z] = 2πi Γ(b1 − s)Γ(b2 − s)Γ(b3 − s) L−∞

# " ¯ ¯ 1 − a1 , 1 − a2 , 1 − a3 Γ(b1 )Γ(b2 )Γ(b3 ) 1,3 ¯ = G −z ¯ . ¯ 0, 1 − b1 , 1 − b2 , 1 − b3 Γ(a1 )Γ(a2 )Γ(a3 ) 3,4

(6.26)

From Theorems 4.1, 5.2 and 5.1, we deduce the following results for the extended generalized hypergeometric function 3 F3 [a1 , a2 , a3 ; b1 , b2 , b3 ; z]:

Theorem 6.7. Let ai , bj ∈ C (i, j = 1, 2, 3) be such that bj 6= − l,

ai + k6= − l

(i = 1, 2, 3; j = 1, 2, 3; k, l ∈ N0 ),

(6.27)

ai + k6=aj + m (i6=j; i, j = 1, 2, 3, k, m ∈ N0 ).

(i) Let z6=0, and let L = L−∞ be the contour which separates all poles cl in (2.1) to the left and all poles ai + k (i = 1, 2, 3; k ∈ N0 ) to the right. Then 3 F3 [a1 , a2 , a3 ; b1 , b2 , b3 ; z] has the series representation 3 F3 [a1 , a2 , a3 ; b1 , b2 , b3 ; z] =

∞ X (a1 )k (a2 )k (a3 )k z k . (b1 )k (b2 )k (b3 )k k!

(6.28)

k=0

(ii)

3 F3 [a1 , a2 , a3 ; b1 , b2 , b3 ; z]

has the asymptotic expansion, as z → 0, ∞ X (a1 )k (a2 )k (a3 )k z k . (b1 )k (b2 )k (b3 )k k!

(6.29)

N X ¡ ¢ (a1 )k (a2 )k (a3 )k z k + O z N +1 (b1 )k (b2 )k (b3 )k k!

(z → 0).(6.30)

3 F3 [a1 , a2 , a3 ; b1 , b2 , b3 ; z]

∼

k=0

In particular, for any N ∈ N0 , 3 F3 [a1 , a2 , a3 ; b1 , b2 , b3 ; z]

=

k=0

Theorem 6.8. Let ai , bj ∈ C (i, j = 1, 2, 3) be such that the conditions in (6.27)

70


are satisfied. Then 3 F3 [a1 , a2 , a3 ; b1 , b2 , b3 ; z] has the asymptotic expansion, as z → → ∞, | arg(−z)| < π/2, µ ¶a Γ(b1 )Γ(b2 )Γ(b3 )Γ(a2 − a1 )Γ(a3 − a1 ) 1 1 F [a , a , a ; b , b , b ; z] ∼ − 3 3 1 2 3 1 2 3 Γ(a2 )Γ(a3 )Γ(b1 − a1 )Γ(b2 − a1 )Γ(b3 − a1 ) z ×

µ ¶k ∞ X (a1 )k (1 + a1 − b1 )k (1 + a1 − b2 )k (1 + a1 − b3 )k 1 − k!(1 + a1 − a2 )k (1 + a1 − a3 )k z

k=0

µ ¶a Γ(b1 )Γ(b2 )Γ(b3 )Γ(a1 − a2 )Γ(a3 − a2 ) 1 2 + − Γ(a1 )Γ(a3 )Γ(b1 − a2 )Γ(b2 − a2 )Γ(b3 − a2 ) z µ ¶k ∞ X 1 (a2 )k (1 + a2 − b1 )k (1 + a2 − b2 )k (1 + a2 − b3 )k × − k!(1 + a2 − a1 )k (1 + a2 − a3 )k z k=0

+

µ ¶a Γ(b1 )Γ(b2 )Γ(b3 )Γ(a1 − a3 )Γ(a2 − a3 ) 1 3 − Γ(a1 )Γ(a2 )Γ(b1 − a3 )Γ(b2 − a3 )Γ(b3 − a3 ) z ×

µ ¶k ∞ X (a3 )k (1 + a3 − b1 )k (1 + a3 − b2 )k (1 + a3 − b3 )k 1 − . k!(1 + a3 − a1 )k (1 + a3 − a2 )k z

(6.31)

k=0

Corollary 6.8.1. Let a1 , a2 , a3 , b1 , b2 ∈ C be such that the conditions in (6.27) are satisfied. Then 3 F3 [a1 , a2 , a3 ; b1 , b2 , b3 ; z] has the following asymptotic estimate, as z → ∞, | arg(−z)| < π/2, 3 F3 [a1 , a2 , a3 ; b1 , b2 , b3 ; z]

µ ¶a ³ ´ Γ(b1 )Γ(b2 )Γ(b3 )Γ(a2 − a1 )Γ(a3 − a1 ) 1 1 = + O z −Re(a1 )−1 − Γ(a2 )Γ(a3 )Γ(b1 − a1 )Γ(b2 − a1 )Γ(b3 − a1 ) z µ ¶a ³ ´ Γ(b1 )Γ(b2 )Γ(b3 )Γ(a1 − a2 )Γ(a3 − a2 ) 1 2 + + O z −Re(a2 )−1 − Γ(a1 )Γ(a3 )Γ(b1 − a2 )Γ(b2 − a2 )Γ(b3 − a2 ) z µ ¶a ´ ³ 1 3 Γ(b1 )Γ(b2 )Γ(b3 )Γ(a1 − a3 )Γ(a2 − a3 ) − + O z −Re(a3 )−1 .(6.32) + Γ(a1 )Γ(a2 )Γ(b1 − a3 )Γ(b2 − a3 )Γ(b3 − a3 ) z In particular, 3 F3 [a1 , a2 , a3 ; b1 , b2 , b3 ; z]

¡ ¢ = O z −ω

(z → ∞),

(6.33)

where ω is given in (6.25). Remark 6.3. When some poles of the gamma functions Γ(ai − s) (i = 1, 2) are not simple (e.g. the conditions in (6.1) and (6.11) are not satisfied), then series representations (1.4), (6.4), (6.12) and asymptotic expansions (6.3), (6.5), (6.13), (6.15) are replaced by the power-logarithmic representations and expansions. Similarly if some poles of the gamma functions Γ(ai − s) (i = 1, 2, 3) are not simple (i.e. the conditions in (6.18) and (6.28) are not satisfied), then series representations (1.5), (6.21), (6.29) and asymptotic expansions (6.19), (6.22), (6.30) and (6.32) are


71

replaced by the power-logarithmic representations and expansions. See Remarks 5.1 and 5.2 in this connection. Acknowledgement The present investigation was supported by the Belarusian Fundamental Research Fund (project F10MC-024), by DGUI of G.A.CC, by MEC, by Science Promotion Fund from the Japan Private School Promotion Foundation and by a grant of University Grants Commission of India. REFERENCES 1.

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