POLYNOMIAL EXPANSIONS FOR SOLUTIONS OF HIGHER-ORDER

POLYNOMIAL EXPANSIONS FOR SOLUTIONS OF HIGHER-ORDER BESSEL HEAT EQUATION IN QUANTUM CALCULUS M.S. Ben Hammouda and Akram Nemri

∗

Abstract In this paper we give the q-analogue of the higher-order Bessel operators studied by I. Dimovski [3],[4], I. Dimovski and V. Kiryakova [5],[6], M. I. Klyuchantsev [17], V. Kiryakova [15], [16], A. Fitouhi, N. H. Mahmoud and S. A. Ould Ahmed Mahmoud [8], and recently by many other authors. Our objective is twofold. First, using the q-Jackson integral and the q-derivative, we aim at establishing some properties of this function with proofs similar to the classical case. Second, our goal is to construct the associated q-Fourier transform and the q-analogue of the theory of the heat polynomials introduced by P. C. Rosenbloom and D. V. Widder [22]. For some value of the vector index, our operator generalizes the q-jα Bessel operator of the second order in [9] and a q-Third operator in [12]. Mathematics Subject Class.: 33C10, 33D60, 26D15, 33D05, 33D15, 33D90 Key Words and Phrases: q-analysis, q-Fourier transform, q-heat equation, q-Laguerre polynomials, q-heat polynomials 1. Introduction The Bessel operator of r-order is defined on (0, ∞) by a1 ar−1 Br u = u(r) + u(r−1) + ... + r−1 u(1) , x x where the coefficients ak depend on the components αk , k αk ≥ −1 + , k = 1, ..., r − 1, r

(1) (2)

40

M.S. Ben Hammouda, A. Nemri

and

k

ar−k

X 1 = (−1)k−j (k − 1)! j=1

µ

j−1 k−1

¶ r−1 Y

(rαi + j),

(3)

i=1

where r is positive integer and α = (α1 , ..., αr−1 ) a vector having (r − 1) components with |α| = α1 + ... + αr−1 . The higher-order Bessel differential operators, called recently as hyperBessel operators, have been introduced by I. Dimovski [3],[4] and studied by I. Dimovski and V. Kiryakova [5],[6], M. I. Klyuchantsev [17], V. Kiryakova [15, Ch. 3], [16], A. Fitouhi, N. H. Mahmoud and S. A. Ould Ahmed Mahmoud [8], and by many other authors (see references in [15], [16]. When r = 2, we obtain the classical Bessel operator of the second order 2α + 1 0 u , (4) x and for r = 3, α1 = −2/3, α2 = ν − 1/3, we obtain the operator B3 u, studied in [1] and in [10] B2 u = u00 +

d3 3 ν d2 3ν d , ν > 0. + − 2 dx3 x dx2 x dx For λ being a complex number, let us now consider the system B3 u =

(5)

Br u(x) = −λr u(x), u(0) = 1, uk (0) = 0,

k = 1, ..., r − 1.

The use of the Frobenius method leads us to conclude that (6) has a unique solution which is r-even and given by µ ¶rm r−1 X 1 Y Γ(αi + 1) λx jα (λx) = (−1)m . (6) m! Γ(αi + m + 1) r m≥0

i=1

In this paper we are concerned with the q-analogue of the jα higherorder Bessel function (6). This choice is motivated in particular by the context of [8], [9], [12]. The reader will notice that the definition (39) derives from that given in [8] with minor changes. With the help of the q-integral representation we establish the q-integral representation of the Mehler and Sonine types. Moreover, we define the higher-order q-Bessel translation and the higherorder q-Bessel Fourier transform and establish some of their properties. Finally, we study the higher-order q-Bessel heat equation.

POLYNOMIAL EXPANSIONS FOR SOLUTIONS OF . . .

41

2. Notation and preliminary results Let q be a fixed real number 0 < q < 1. We use the following notation: n−1 Y (a + b)nq = (a + q j b), if n = 0, 1, 2, ..., ∞, (7) j=0

(1 + a)tq =

(1 + a)∞ q , (1 + q t a)∞ q

if t ∈ C,

(8)

and put • Rq = {±q k , k ∈ Z} ∪ {0},

Rq,+ = {q k , k ∈ Z} and (n)2 =

Note that for λ ∈ R, n = 0, 1, 2, ..., (a; q)n = (1 − a)(1 − aq)...(1 − aq n−1 ), (λ)qn = (λ)qn (λ + n − 1)q

(q λ ; q) , (1 − q)n

= (λ)qn−1 ,

[n]q ! =

(q; q)n , (1 − q)n

n(n−1) . 2

1 − qλ , 1−q (q λ ; q)n (λ)qn = , [n]q ! (q; q)n (λ)q =

(1)qn = (−1)k (−n)qk q nk−(k)2 . q (1)n−k

and then, the q-Binomial formula is: ¸ n · X n (ab; q)n = bk (a; q)k (b; q)n−k , k q

·

¸

(q; q)n . (q; q)n−k (q; q)k q k=0 (9) Further we denote by Dq the q–derivative of a function by: f (qx) − f (x) Dq f (x) = . (10) (q − 1)x · ¸ n q −(n)2 X n n k Dq f (x) = n (−1) q (n−k)(n−k)/2 f (q k x), n = 0, 1, 2, .... k q x (1 − q)n with

n k

=

k=0

Dqn [f (x)g(x)] =

(11)

¸ n · X n (Dn−k f )(q k x)(Dqk g)(x), n = 0, 1, 2, ..., k q q k=0

and define the q-shift operators by: (Λq f )(x) = f (qx) noting that (Λ−1 f )(x) = f (q −δ x). qδ

and

−1 (Λ−1 q f )(x) = f (q x),

(12)

42


The q-Jackson integrals (introduced by Thomae and Jackson [13]) from 0 to a and from aq to ∞ are defined by Z

a

0

Z ∞ X j j f (x)dq x = (1−q) aq f (aq ) and

∞

aq

j=0

+∞ X f (t)dq t = (1−q) aq −k f (aq −k ). k=0

(13) Notice that the last series are guaranteed to be convergent, see [9]. We define the Jackson integral in a generic interval [a, b] by [13]: Z b Z b Z a f (x)dq x = f (x)dq x − f (x)dq x . a

0

0

This is a special case of the following more general change of variable formula, [14, p 107]. If u(x) = αxβ , then Z u(b) Z b f (u)dq u = f (u(x))Dq1/β u(x)dq1/β x . u(a)

Z

1

0

a

Using Z 1 the q-Jackson integrals from 0 to 1, we define the q-integral ... f (t1 , ..., tn )dq t1 ...dq tn by: 0

Z

Z

1

... 0

0

1

f (t1 , ..., tn ) dq t1 ...dq tn = (1 − q)n

∞ X

q i1 +...+in f (q i1 +...+in ),

i1 ,...,in =0

(14) provided the sums converge absolutely. We present two q–analogues exponential function: Eq (x) = eq (x) =

∞ X n=0 ∞ X n=0

q (n)

xn = (1 + (1 − q)x)∞ q , [n]q !

xn 1 = . [n]q ! (1 − (1 − q)x)∞ q

(15) (16)

Notice that for q ∈ (0, 1) the series expansion of eq (x) has radius of convergence 1/(1 − q). On the contrary, the series expansion of Eq (x) converges for every x. Both product expansions (15) and (16) converge for all x. We define the q δ -basic hypergeometric series r φδs by ¯ µ a ¶ X ∞ q q k q 1 , ..., q ar ¯¯ δ 1+s−r δ (k)2 (a1 ; q)k ...(ar ; q)k z q; (q − 1) z = (q ) , r φs q q q b1 , ..., q bs ¯ (b1 ; q)k ...(bs ; q)k [k]q ! k=0 (17)

POLYNOMIAL EXPANSIONS FOR SOLUTIONS OF . . . µ lim r φδs q↑1

q a1 , ..., q ar q b1 , ..., q bs

¯ ¶ · ¯ ¯q; (q − 1)1+s−r z =r Fs a1 , ...ar ¯ b1 , ...bs

¯ ¸ ¯ ¯z . ¯

43

(18)

Here δ > 0 and r < s + 1, thus the expansion converges for all values of z. For δ = 1 + s − r, we obtain the classical basic hypergeometric series r φs , [18, p 11,12]. ∞ X xn Note that for δ > 0 by eq (x, δ) = , this expansion conq δ(n)2 [n]q ! n=0 verges for all values of x. The q–gamma function Γq (t), a q–analogue of Euler’s gamma function, was introduced by Thomae and later by Jackson as the infinite product (1 − q)t−1 q Γq (t) = (19) , t>0. (1 − q)t−1 The q–Beta function defined by the usual formula Γq (s)Γq (t) βq (t, s) = , (20) Γq (s + t) has a q–integral representation, which is a q–analogue of Euler’s formula: Z 1 βq (t, s) = xt−1 (1 − qx)s−1 t, s > 0 . (21) q dq x , 0

The q-duplication formula holds: r−1 r−1 Y Y i 1 i 1 1 Γqr (n + ) = Γq r ( ) , r [rn]q ! r [n]qr ! ((r)q )rn i=1

and

(22)

i=1

r

((r)q )rn (1)qn

r−1 Y i=1

We also denote,

i r ( )qn = [rn]q !. r

(23)

r−1 Y Y r r (αi + 1)qn = (αi + 1)qn . i=1

3. q-Trigonometric function of r-order The r − q δ -cosinus is defined for δ > 0 by µ r

cosr (x, q ; δ) =

δ 0 φr−1

− (q r )1/r , ..., (q r )(r−1)/r

=

X m≥0

¯ ¶ ¯ r r r r ¯q ; − (q −1) xr−1 r (24) ¯ (1+q+...+q )

(−1)m brm (x, q r ; δ),

(25)

44


where brm (x, q r ; δ) = (q δ )r(m)2

xrm xrm . = (q r )δ(m)2 [rm]q ! αrm,q

(26)

For every λ ∈ C, the function cosr (x, q r ; δ) is a unique solution of the system  −1 r  Λqδ Dq u(x) = −λr u(x), u(0) = 1,  k Dq u(0) = 0, k = 1, ..., r − 1. We note r − q δ -sinus of order (r, l) , l = 1, ..., r − 1 by sinr,l (x, q r ; δ) =

X

(−1)m (q δ )r(m)2

m≥0

xrm+r−l . [rm + r − l]q !

Let µ = eiπ/r and wk = e2iπ(k−1)/r , k = 1, 2, ..., r. Since ½ r X r for integers m divisible by r m (wk ) = 0 for integers m not divisible by r

(27)

(28)

k=1

and expanding the q-exponential function in series, we obtain cosr (x, q r ; rδ) =

r ¢ 1 X ¡ µwk x eqr (r−1)/2 , δ . r q

(29)

k=1

When r = 3, δ = 1, we obtain the result in [12]. Definition 3.1. Let x ∈ R and wk = e2iπ(k−1)/r , k = 1, 2, ..., r, a function f (x) is called r-even, if f (wk x) = f (x)

k = 1, ..., r,

(30)

and r-odd of l order, if f (x) = wkl f (wk x),

k = 1, ..., r.

(31)

Proposition 3.1. The functions cosr and sinr,l (l = 1, ..., r − 1) are, respectively, r-even and r-odd of order l. From (24) and (27) we obtain the following q-derivative formulas: Dql cosr (x, q r ; δ) = − q −δ(r−l) sinr,l (q δ x, q r ; δ), Dqr cosr (x, q r ; δ) = − cosr (q δ x, q r ; δ, ), Dql−m sinr,m (x, q r ; δ) = sinr,l (x, q r ; δ), Dqr−m sinr,m (x, q r ; δ) = cosr (x, q r ; δ).


45

Proposition 3.2. The function cosr (x, q r ; 1) is r-even and satisfies, in particular cosr (xt, q r ; 1) = (−1)n q r(n(n+1)/2)

1 rn D (cosr (xtq −n , q r ; 1)). xrn q,t

(32)

Proposition 3.3. Let x ∈ R for n ≥ 1, the function brn (x, q r ; 1) verifies the following properties r r r b0 (x, q r ; 1) = 1, brn (0, q r ; 1) = 0 and Λ−1 q Dq brn (x, q ; 1) = br(n−1) (x, q ; 1).

Furthermore, | brn (x, q r ; δ) |≤| brn (x, q r ; 1) |≤ P r o o f. When we put q = e−t , defined by ( 24) can be written as r

brn (x, q ; 1) =

n−1 Y r−1 Y j=0 i=0

q −(r)2 xrn , δ ≥ 1. (rn)!

(33)

t > 0. The coefficients brn (x, q r ; 1)

n−1 Y e−jt − e−(j+1)t Y r−1 q j − q j+1 . = 1 − q rj+1+i 1 − e−(rj+1+i)t j=0 i=0

Preceding like in [20], we can deduce the result and we have when |x| ↑ ∞, we have | cosr (x, q r ; δ)| ≤ q −(r)2 | cosr (x)| ≤ q −(r)2 e(r−2)|x| , δ ≥ 1, see [9]. 4. q δ -product formula We set now the product formula for q δ -cosinus function. We note by P = cosr (x, q r ; δ) cosr (y, q r ; δ). Proposition 4.1. Let x and y be complex numbers, with y 6= 0, we have: · ¸ rk X (−1)k (qδ )rk2 ¡ x ¢rk X rk s (s) −(rk) 2 2 (−1) q Λ−k cosr (yq rk−s , q r ; δ). P= q y (1−q)rk [rk]q ! qδ s q s=0

k≥0

P r o o f. For y 6= 0, P =

X (q δ )rk2 ¡ x ¢rk X k≥0

[rk]q !

y

n≥0

(−1)n

(q rδ )(n)2 (q δ )−rnk y rn . [r(n − k)]q !

46


Moreover, if we use the previous relation rk

X [rn]q ! 2 (1 − q)rk = (−1)rk q −(rk)2 +r nk (−1)s q (s)2 [r(n − k)]q ! s=0

·

rk s

¸ q −rns , q

we obtain that · ¸ rk X (−1)k q−(rk)2 (qδ )rk2 ¡ ¢rk X rk x s (s) 2 P= (−1) q cosr (yq k(r−δ) q −s , q r ; δ). y (1−q)rk [rk]q ! s q k≥0

s=0

5. The q-Bessel operator of r-order We suppose now that the components of the vector α = (α1 , ..., αr−1 ), where αk is a real number, satisfy αk ≥ −1 + kr , k = 1, ..., r − 1 and δ > 0. The q-Bessel operator of r-order is defined by: Y¡ ¢ ¢ ¡ 1 r−1 q rαi +1 xDq + (rαi + 1)q Dq u . Br,δ u = Λ−1 q δ xr−1

(34)

i=1

Remark 5.1. For r = 2, we obtain the q-Bessel operator B2,δ of the second order studied in [9] for δ = 1: ¡ 2α+1 2 ¢ (2α + 1)q B2,δ u = Λ−1 q Dq u + Dq u (35) qδ x and for r = 3, α1 = −2/3, α2 = ν −1/3, we obtain the operator B3,δ studied in [12]: ¡ 3ν 3 ¢ 1 (3 ν)q 2 1 (3 ν)q B3,δ u = Λ−1 q D u + D u − D u . (36) q δ q q 2 q q x q x Proposition 5.1. For λ in C, the function jα (λx, q r , δ) ¯ µ ¶ ¯ r − λr (q r −1)r xr r δ ¯ q ; − (1+q+...+qr−1 )r jα (λx, q , δ) = 0 φr−1 (37) (q r )α1 +1 , ..., (q r )αr−1 +1 ¯ is a solution of the q-problem   Br,δ u(x) = −λr u(x) u(0) = 1,  k Dq u(0) = 0, k = 1, ..., r − 1 .

(38)

POLYNOMIAL EXPANSIONS FOR SOLUTIONS OF . . . Furthermore, jα (λx, q r , δ) has the following representation ∞ X jα (λx, q r , δ) = (−1)n brn,α (x, q r , δ)λrn ,

47

(39)

n=0

where brn,α (x, q r , δ) = and

(q r )δ(n)2 xrn (q r )δ(n)2 xrn = , Q r r αrn,α,q ((r)q )rn (1)qn (αi + 1)qn

αrn,α,q = (1 + q + ... + q

r−1 rn

)

[n] !

r−1 Y

qr

i=1

Γqr (αi + n + 1) . Γqr (αi + 1)

(40)

(41)

For δ = r, we obtain the q-hypergeometric function 0 φr−1 . Let now | α |= α1 + .... + αr−1 = α0 + .... + αr−1 with α0 = 0, brn,α (1, q r ; δ) ≤ brn,α (1, q r , 1) =

(q r )(n)2 rQ r, ((r)q )rn (1)qn (αi + 1)qn

the right term can be written by ¡ r −|α|/r ¢n n−1 r−1 Y Y (q r )(αi +j)/r − (q r )1+(αi +j)/r (q ) . ((r)q )rn 1 − (q r )1+(αi +j)

δ ≥ 1,

(42)

(43)

j=0 i=0

Now, by [19], Lemma A.1, [20] and Proposition A.2, we see that the general terms of product increases to (j + αi + 1)−1 if q ↑ 1. Using Stirling’s formula, we find that, for some constant C, ¡ r −|α|/r ¢n ¶rn+|α| µ ¡ r −|α|/r ¢n (q ) e r Q brn,α (1, q ; δ) ≤ ≤ C (q ) , ((r)q )rn (αi + 1)n n(r)q (44) this inequality generalizes the inequality in [12]. Proposition 5.2. For αi ≥ −1 + ri , i = 1, ..., r − 1, and n = 0, 1, 2..., µ ¶ x r−1 1 r Q Dq jα (., q , δ)(x) = − jα+1 (q δ x, q r , δ), (45) (r)q (αi + 1)qr and

µ ¶ ¡ 1 ¢r−1 n (−1)n (q δ )(n)2 nδ r { r−1 Dq } jα (x, q , δ) = Q q r jα+n (q x, q , δ). x (r)q (αi + 1)n (46) By the q-duplication formula of Γq (22), we have in particular 1

n

r

j(−1/r,−2/r,...,−(r−1)/r) (x, q r , δ) = cosr (x, q r ; δ).

(47)

48

M.S. Ben Hammouda, A. Nemri 6. q-integral representations

In this section, we give two q-integral representations of the q-jα function (39) involving the q-Jackson integral. We denote by Wα the function r−1 Y

r

Wα (t1 , ..., tr−1 ; q ) = which tends to

(tri q r ; q r )∞

r αi − ri +1 ; i=1 (ti q

q r )∞

tii−1

=

r−1 Y

(tri q r ; q r )αi − i tii−1 r

i=1

Q i (1 − tri )αi − r ti−1 as q −→ 1− . i

Theorem 6.1. For αi ≥ −1 + ri , i = 1, ..., r − 1, the function jα has the following q-integral representation of Mehler type Z 1 Z 1 jα (z, q r , δ) = Cr,α ... Wα (t1 , ..tr−1 ; q r ) cosr (zt1 , ...tr−1 ; q r , δ) dq t1 ..dq tr−1 , 0

0

(48)

where r−1

Cr,α = ((r)q )

r−1 Y

Γqr (αi + 1) i Γ r ( )Γqr (αi − ri i=1 q r

+ 1)

.

(49)

P r o o f. This formula can be proved by expanding cosr (zt, q r ; δ) in a series of power of t and then there arise q-integrals of the form Z 1 i Γqr (m + ri )Γqr (αi − ri + 1) r r αi − r i−1 ti d q ti = . (50) trm i (1 − q ti )q r (r)q Γqr (αi + m + 1) 0 Based on the q-duplication formula for the Γq function (22), the formula is proved. Proposition 6.1. For αi ≥ −1 + ri , i = 1, ..., r − 1, and n = 0, 1, 2..., ¯ ¯ ¯ ¯ Y Γqr (αi + 1)Γqr ( n+i ) ¯£ ¯ n£ ¤¯ r−1 ¤¯ n r r ¯Dq jα (x, q r , δ) ¯ ≤ ¯ Dq,x cosr (x, q ; δ) ¯¯, (51) ¯ ¯ i n ¯ r ( )Γq r (αi + 1 + Γ ) q r r i=1 in particular

¯ ¯ ¯jα (x, q r , δ)¯ ≤ q −(r)2 e(r−2)|x| .

(52)

Theorem 6.2. For αi ≥ −1 + ri , i = 1, ..., r − 1 and pi ≥ 1, the function jα+p has the following q-integral representation of Sonine type Z 1 Z 1 r p jα+p (z, q , δ) = cr,α ... Vp (t1 , ..tr−1 ; q r ) jα (zt1 , ..tr−1 , q r ; δ) dq t1 ...dq tr−1 , 0

0

(53)

POLYNOMIAL EXPANSIONS FOR SOLUTIONS OF . . . where cpr,α

r−1

= ((r)q )

Vp (t1 , ..., tr−1 ; q r ) =

r−1 Y

i=1 r−1 Y

Γqr (αi + pi + 1) Γqr (pi )Γqr (αi + 1) r(αi − ri +1) i−1 ti

(1 − q r tri )pqri ti

49

(54) .

(55)

i=1

P r o o f. This formula can be proved by expanding jα in a series of power of ti , there arise q-integrals of the form Z 1 Γqr (αi + m + 1)Γqr (pi ) r(αi − ri +1+m) ti (1 − q r tri )pqri −1 ti−1 d q ti = . i (r)q Γqr (m + αi + pi + 1) 0 7. q-Fourier transform Notations: Some q-functional spaces will be used to establish our result. We design by E∗,q (R) (resp. E∗,q (Rq )) the space of r-even functions defined on R (resp Rq ) infinitely q-derivative, and by D∗,q (R) (resp D∗,q (Rq )) the space of r-even functions defined on R (resp. Rq ) infinitely q-derivative with compact support. In this section we introduce the space L1α,qδ (Rq,+ , dq x) of functions f satisfying Z ∞

0

|f (x)jα (λx, q r ; δ)| dq x < ∞,

λ ∈ Rq .

Definition 7.1. The Fourier transform related with Br,δ of f ∈ L1α, qδ (Rq,+ , dq x) is the function Z ∞ Fqδ (f ) defined by Fqδ (f )(λ) = f (t) jα (λt, q r ; δ) dq t, λ ∈ Rq . (56) 0

We define also the FourierZtransform F0,qδ by ∞ F0,qδ (f )(λ) = f (t) cosr (λt, q r ; δ) dq t, 0

λ ∈ Rq .

(57)

8. q-translation and q-convolution In this section we study the generalized translation operator associated r with the operator Br,δ . We give the following definition related to Λ−1 qδ Dq .

50


Definition 8.1. The translation operator τx, qδ , x ∈ R (resp Rq ) r associated with the r-order derivative operator Λ−1 qδ Dq is defined for f in E∗,q (R) (resp. E∗, q (Rq )) and y ∈ R (resp. Rq ) by τx, qδ (f )(y) =

∞ X

r (n) f (x), brn (y, q r ; δ) (Λ−1 qδ Dq )

(58)

n=0

the functions brn (y, q r ; δ) are given by (24). Proposition 8.1. The operators τx, qδ satisfy: 1- the product formula: cosr (λx, q r ; δ). cosr (λy, q r ; δ) = τx, qδ cosr (λy, q r ; δ) = τy, qδ cosr (λx, q r ; δ). 2- For x ∈ R, τx, qδ belong in L(E∗,q (R), E∗,q (R)). 3- The map x −→ τx, qδ is infinitely q-derivative, r-even. Lemma 8.1. For f ∈ D∗,q (R), n ∈ N , we have: ¡ −1 r ¢n Λqδ Dq f (x) =

rn

X (−1)k q (rn−k)2 q −(rn)2 Λ−n f (q k x). [rn − k]q ![k]q ! qδ (1−q)rn (q −δn )rn brn (x, q r ; δ) k=0

P r o o f. For δ > 0, by [21] and relation (11), · ¸ rn X q −(rn)2 rn k rn (−1) Dq,x f (x) = q (rn−k)2 f (q k x), k (1 − q)rn xrn q k=0

¡ ¢n ¡ ¢−(n)2 −n rn using the fact that Λ−1 Dqr = (q δ )r Λqδ Dq , then we obtain qδ ¡ ¢−(n)2 rn · ¸ X ¡ −1 r ¢n q −(rn)2 (q δ )r rn k (−1) Λqδ Dq f (x) = q (rn−k)2 Λ−n f (q k x). qδ k (1 − q)rn (q −δn x)rn q k=0

this leads to the result. Remark 8.1. We obtain for δ > 0 τy,qδ f (x) =

∞ X b

r −(rn)2 rn (1, q ;δ)q (1−q)rn (q −δn )rn

n=0

rn ¡ y ¢rn X x k=0

· (−1)k

rn k

¸ q

q (rn−k)2 Λ−n f (q k x). qδ

Proposition 8.2. For f ∈ D∗,q (Rq ) we have: F0,qδ (t τx, qδ f )(λ) = cosr (λx, q 3 ; δ)F0,qδ (f )(λ),

(59)


51

the convolution product of two functions f and g of D∗,q (Rq ) is defined in D∗, q (Rq ) by: Z ∞ Z ∞ t (60) τx, qδ f (y)g(y) dq y = f (y)τx, qδ g(y) dq y, f ?qδ g(x) = 0

0

and we have: F0, qδ (f ?qδ g)(λ) = F0, qδ (f )(λ). F0, qδ (g)(λ).

(61)

Definition 8.2. We call generalized translation operators associated with Br,δ , the operators Tαx, qδ , x ∈ R (resp. Rq ), defined on E∗, q (R) (resp. E∗,q (Rq )) by: Tαx, qδ (f )(y)

=

∞ X

n brn, α (y, q r ; δ) Br,δ (f )(y),

y ∈ R (resp Rq ),

(62)

n=0

where the functions brn,α (y, q r , δ) is given by (39). Proposition 8.3. The operators Tαx, qδ satisfy: 1. For x ∈ R, Tαx,qδ in L(E∗,q (R), E∗,q (R)). 2. The map x −→ Tαx,qδ are infinitely q-derivative and r-even. 3. For all functions f in E∗,q (R): – Tαx,qδ f (y) = Tαy,qδ f (x) – Tα0,qδ f (y) = f (y). 4. For given f in E∗,q (R), we put: u(x; y) = Tαx,qδ f (y), then the function u is solution of the Cauchy problem:   Bx,r,δ u(x, y) = By,r,δ u(x, y), u(x, 0) = f (x); Dq, y u(x, 0) = 0 (I)  k Dq,y u(x, 0) = 0 , k = 0.1.2...r − 1 and we have Tαx, qδ jα (λy, q r , δ) = jα (λx, q r , δ)jα (λy, q r , δ) = Tαy, qδ jα (λx, q r , δ).

(63)

Now we are able to define the convolution product related to the operator Br,δ .

52


Definition 8.3. The convolution product associated with Br,δ of two functions f and g in D∗, q (Rq ) is the function f ?α, qδ g defined by: Z f ?α, qδ g(y) =

∞

0

Z f (x)Tαy, qδ g(x) dq x

∞

= 0

t

Tαy, qδ f (x) g(x) dq x.

(64)

9. Higher-order q-Bessel heat polynomials We recall that the function eqr (−z r t)jα (xz; q r ; δ) is analytic in z r . We thus have, for t ∈ R and δ ≥ 1, r

r

eqr (−z t)jα (xz; q ; δ) =

∞ X

(−1)n

n=0

then pαn (x,

z rn αrn,α,q

pαn (x, t, q r ; δ),

n X (xr )n−k tk αrn,α,q (q r )δ(n−k)2 t, q ; δ) = [k]qr ! αr(n−k),α,q r

(65)

(66)

k=0

= =

r Q r ∞ X (−1)k (−n)qk (q r )(δ−1)(k)2 (q r )nk (xr )k t−k (αi + 1)qn n t Q r (1 + q + ... + q r−1 )−rn (αi + 1)qk (1 + q + ... + q r−1 )rk [k]qr ! k=0 ¯ µ ¶ ¯ r (qr −1)r−1 (−xr (qr )n ) αrn,α,q n δ−1 (q r )−n ¯ t 1 φr−1 . q ; (1+q+..+qr−1 )r t (q r )α1 +1 , .., (q r )αr−1 +1 ¯ [n]qr !

10. Application: q-heat equation We give an applications of the Fourier transform related with Br,δ . We begin by recalling that Z 0

r

∞

eqr (−cxr ) (cn xrn )xrαk +(r−1) dq x =

(q r )−n(αk +1)−(n)2 (αk + 1)qn I (αk ,+1,qr ) cαk +1 (1 + q + ... + q r−1 ) (67)

where Z r

I(αk + 1; q ) =

0

∞

eqr (−x) xαk dqr x and Hqr (αk + 1) =

I(αk + 1; q r ) . Γqr (αk + 1) (68)


53

rαk +(r−1)

y We note by dηq,αk (y) = (1+q+...+q r−1 )αk Γ r (α +1) dq y and we define for δ > 1 k q the fundamental solution Kαk (x, t, q r ; δ) by

Z r

Kαk (x, t, q ; δ) = =

Hqr (αk +1) (t(1+q+...+q r−1 ))αk +1

=

Hqr (αk +1) (t(1+q+..q r−1 ))αk +1

×

δ−1 B 0 φr−2 B @

0

0

∞

eqr (−ty r )jα (xy, q r ; δ)dηq,αk (y),

∞ X (−1)n (q δ )r(n)2 × (q r )−(αk +1)n−(n)2 xrn t−n Q qr , r−1 )rn [n] r ! (1 + q + ... + q (α + 1) n q i i6 = k n=0

− (q r )α1 +1 , .., (q r )αr−1 +1

¯ 1 ¯ r −xr (qr )−(αk +1) (q r − 1)r−1 C ¯q ; ( )C A. ¯ (1+q+..q r−1 )r t

For δ = r, we obtain the basic hypergeometric series. We consider the q-problem for t, x ≥ 0  Br,δ u(x, t) = Dqr , t u(x, t)    k Dq u(0, t) = 0, k = 1, ..., r − 1 (II) u(wk x, t) = u(x, t), k = 1, ..., r − 1    u(x, 0) = f (x). Theorem 10.1. Let f ∈ L1α, qδ (Rq,+ , dq x), the function Z u(x, t) = 0

∞

¡ ¢ Tαy, qδ Kαk (x, t, q r ; δ) f (y) dq y = f ?α,qδ Kαk (. , t, q r ; δ) (x),

(69) is a solution of the equation (II) for αk ≥ −1+ kr , k = 1, ..., r−1, t, x ∈ Rq,+ . 11. Analytic Cauchy problem related to the r-order q-Bessel operator Br,δ We say that a function u(x, t) in Hα ([0, a] × [0, σ]) if Br,δ u(x, t) = Dqr ,t u(x, t).

(70)

The diffusion polynomials pαn (x, t) satisfy the q-equation (70). Hence we P expect to obtain infinite series expansions u(x, t) = m≥0 am pαm (x, t, q r ; δ) with possible convergence in a strip |t| < σ.

54

M.S. Ben Hammouda, A. Nemri Let δ ≥ 1, we note ∞ X

(q r )(δ−1)(n)2 xrn Q r (1 + q + ... + q r−1 )rn (αi + 1)qn n=0 ¯ ¶ µ r 1 ¯ r (qr −1)r−1 xr (q ) δ−1 ¯ q ; . = 1 ϕr−1 (q r )α1 +1 , ..., (q r )αr−1 +1 ¯ (1+q+..+qr−1 )r

Rδα,q (x) =

Lemma 10.1. Let s > 0 and δ ≥ 1 ¡ |x| ¢ pαn (|x|, |t|, q r , δ) sn |t| ≤ (1 + )n Rδα,q 1/r . αrn,α,q [n]qr ! s s P r o o f. We have pαn (|x|, |t|, q r , δ) αrn,α,q

∞

≤

sn X [n]qr !

·

k=0

≤ Rδα,q ( = ≤

n k

¸ µ qr

|t| s

¶n−k

∞ |x| sn X r (k)2 (q ) ) s1/r [n]qr ! k=0

sn ¡ 1+ [n]qr ! sn ¡ 1+ [n]qr !

rk

(q r )δ(k)2 |x|sk Q r ((r)q )rk (αi + 1)qn · ¸ µ ¶n−k |t| n k qr s

|t| ¢n δ |x| r Rα,q ( 1/r ) q s s ¢ |t| n δ |x| Rα,q ( 1/r ), s s

rk

since,

(q r )δ−1(k)2 |x|sk |x| δ Q q r < Rα,q ( 1/r ). rk s ((r)q ) (αi + 1)n

Lemma 10.2. For t, x > 0, δ > 0 pαn (x, t, q r , δ) ≥

αrn,α,q n t . [n]qr !

P r o o f. Since the coefficients of pαn are positive, it follows that pαn (x, t, q r , δ) ≥ pαn (0, t, q r , δ) =

αrn,α,q n t . [n]qr !


Theorem 10.2. If the series and x0 > 0, the series

∞ X

∞ X

55

an pαn (x0 , t0 , q r , δ) converges for t0 > 0

n=0

an pαn (x0 , t0 , q r , δ) and

n=0

∞ X

drn,α,q an pαn−1 (x, t, q r , δ)

n=0

converge absolutely and locally uniformly in the strip |t| < t0 and the series ∞ X an pαn (x0 , t0 , q r , δ) is in Hα (R+ ) for |t| < t0 . n=0

P r o o f. We note by drn,α,q = αrn,α,q /αr(n−1),α,q . Since the general term of a convergent series must go to zero, lim an pαn (x, t, q r , δ) = 0. By n−→∞ ¡ [n]qr ! ¢ . Using Lemma Lemma 10.2, it therefore follows that an = O αrn,α,q tn0 10.1, we get for s > 0 and δ ≥ 1 ∞ X

an drn,α,q pαn−1 (x, t, q r , δ) ≤ M

n=0

∞ X n=1

[n]qr ! αrn,α,q |x| (s + |t|)n Rδα,q ( 1/r ) n αrn,α,q t0 [n]qr ! s ∞ |x| X ¡ s + |t| ¢n ) t0 s1/r n=0

≤ M Rδα,q (

which converges for s + |t| < t0 . Since s > 0 is arbitrary it converges for (s + |t|) < t0 , and as before for |t| < t0 . ∞ X Let f (x) = an xn be an entire function of order ρ, ρ > 0, and of type n=0

0 < σ < ∞. The type is determined by lim sup n→∞

|an | ≤ M

³ eσρ ´rn/ρ rn

ρ rn |an | rn = σ. Therefore, eρ

.

(71)

Theorem 10.3. If f (z) is an entire function of order ρ with 0 < ρ < r/r − 1 and of type σ, 0 < σ < ∞, then u(x, t) =

∞ X

an pαn (x, t, q r , δ)

(72)

n=0

is in Hα (R) in the strip |t| < 1/(σρ)r/ρ and u(x, 0) = f (x). P r o o f. Using (71) and Lemma 10.1, for s > 0 we obtain ∞ ³ X |x| eσρ ´rn/ρ αrn,α,q α r an pn (x, t, q , δ) ≤ M (s + |t|)n Rδ α,q ( 1/r ). (73) rn [n]qr ! s

∞ X

n=0

n=0

56

M.S. Ben Hammouda, A. Nemri r−1 ³ eσρ ´rn/ρ α ³ eσρ ´rn/ρ Y Γqr (αi + n + 1) rn,α,q rn Since, ≤ r , or for rn [n]qr ! rn Γqr (αi + 1)

n ↑ ∞, we have for n ↑ ∞ we get

r−1 Y

i=1

Γ (αi + n + 1) ∼ Γqr (αi + 1) qr

Γqr (αi + n + 1) ≤

r−1 Y

i=1

³ eσρ ´rn/ρ rn

r−1 Y

rrn

for 0 < ρ
0 is arbitrary, we get 1 absolute and local uniform convergence for |t| < . Since the order (σρ)3/ρ and type of entire function is not changed by Ptaking derivatives, a similar type argument shows that the derived series n≥1 an drn,α,q pαn−1 (x, t, q r , δ), 1 also converges absolutely and locally uniformly for |t| < . It follows (σρ)r/ρ that u(x, t) given by (72) is in Hα in the stated strip. References [1] F. M. Cholewinski and D. T. Haimo, Classical analysis and the generalized heat equation. SIAM Review 10 (1968), 67-80. [2] F. M. Cholewinski and J. A. Reneke, The generalized Airy diffusion equation. Electronic Journal of Differential Equations 87 (2003), 1-64. [3] I. Dimovski, Operational calculus for a class of differential operators. C.R. Acad. Bulg. Sci. 19 (1966), 1111-1114. [4] I. Dimovski, Foundations of operational calculi for the Bessel-type differential operators. Serdica. Bulg. Math. Publ-s 1(1975), 51-63.


57

[5] I. Dimovski, V. Kiryakova, On an integral transformation, due to N. Obrechkoff. Lecture Notes in Math. 798(1980), 141-147. [6] I. Dimovski, V. Kiryakova, The Obrechkoff integral transform: properties and relation to a generalized fractional calculus. Numerical Functional Analysis and Optimization 21, No 1-2 (2000), 121-144. [7] A. Fitouhi, Heat polynomials for a singular operator on (0,∞). Constr. Approx. 5 (1989), 241-270. [8] A. Fitouhi, N. H. Mahmoud and S.A. Ould Ahmed Mahmoud, Polynomial expansions for solutions of higher-order Bessel heat equations. JMAA 206(1997), 155-167. [9] A. Fitouhi, M. M. Hamsa and F. Bouzeffour, The q − Jα Bessel function. Journal of Approximation Theory 115(2002), 144-166. [10] A. Fitouhi, M. S. Ben Hammouda and W. Binous, On a third singular differetial operator and transmutation. Far East J.Math.Sci.(FJMS) (2005), Reference No 051003. [11] D. T. Haimo, L2 expansions in terms of generalized heat polynomials and of their Appell transforms. Pacific J. Math 15(1965), 865-875. [12] M. S. Ben Hammouda and A. Fitouhi, Polynomial expansions for solutions of a third q-heat equation. To appear. [13] F. H. Jackson, On q–definite integrals. Quart. JMPA 41(1910), 193203. [14] V. Kac and P. Cheung, Quantum Calculus. Universitext, SpringerVerlag, New York (2002). [15] V. Kiryakova, Generalized Fractional Calculus and Applications. Longman, Harlow; John Wiley, N. York (1994). [16] V. Kiryakova, Obrechkoff integral transform and hyper-Bessel operators via G-function and fractional calculus approach. Global Jouranl of Pure and Applied Mathematics 1, No 3 (2005), 321-341. [17] M. I. Klyuchantsev, Singular differential operators with r − 1 parameters and Bessel functions of vector index. Seberian Math. J. 24(1983), 353-366.

58


[18] R. Koekoek and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Report 98-17. [19] T. H. Koornwinder, Jacobi functions as limit cases of q-ultraspherical polynomials. Journal of Mathematical Analysis and Applications 148 (1990), 44-54. [20] T. H. Koornwinder and R. F. Swarttouw, On q-Analogue of the Fourier and Hankel transforms. Trans. Amer. Math. Soc. 333(1992), 445-461. [21] T. H. Koornwinder, Some simple applications and variants of the q-binomial formula. Downloadable from: www.science.uva.nl/pub/mathematics/reports/Analysis/qbinomial.ps, 1999. [22] P. C. Rosenbloom and D. V. Widder, Expansions in terms of heat polynomials and associated functions. Trans. Amer. Math. Soc. 92(1959), 220-266. [23] A. De Sole and Victor Kac, On integral representations of q-gamma and q-beta functions. arXiv: math. QA/0302032, 2003. [24] K. Trimèche, Transformation integrale de Weyl et théorème de PaleyWiener associés un opérateur differentiel singulier sur (0, ∞). JMPA 60(1981), 51-98.

Département de Matématiques Faculté des Sciences de Tunis Campus Universitaire Tunis 1060, TUNISIA

Received: March 10, 2007 Revised: April 16, 2007

e-mail: [email protected] ∗ e-mail: [email protected] (corr. author)