ON CERTAIN CHARACTERIZATIONS AND INTEGRAL

ˇ GLASNIK MATEMATICKI Vol. 37(57)(2002), 93 – 100

ON CERTAIN CHARACTERIZATIONS AND INTEGRAL REPRESENTATIONS OF CHATTERJEA’S GENERALIZED BESSEL POLYNOMIAL Mumtaz Ahmad Khan and Khursheed Ahmad Aligarh Muslim University, India

Abstract. The present paper deals with certain recurrence relations, integral representations, characterizations and a Rodrigue’s type n-th derivative formula for the generalized Bessel polynomial of Chatterjea.

1. Introduction In 1949 Krall and Frink [10] initiated serious study of what they called Bessel polynomials. In their terminology the simple Bessel polynomial is h xi (1.1) yn (x) = 2 F0 −n, 1 + n; −; − 2 and the generalized one is h xi (1.2) yn (a, b, x) = 2 F0 −n, a − 1 + n; −; − . b

Several other authors including Agarwal [1], Al-Salam [2], Brafman [3], Burchnall [4], Carlitz [5], Dickinson [8], Grosswald [9], Rainville [11] and Toscano [14] have contributed to the study of the Bessel polynomials. In 1965, Chatterjea [7] generalized (1.2) and obtained certain generating functions for his generalized polynomial defined by 2 F0

(−n, C + kn; −; x) ,

where k is a positive integer. 2000 Mathematics Subject Classification. 33A65. Key words and phrases. Generalized Bessel polynomials, recurrence relations, integral representations. 93

94

M. A. KHAN AND K. AHMAD

In this paper we propose to study further the generalized Bessel polynomials due to Chatterjea [7]. In particular we shall find some recurrence relations, integral representations and certain characterizations of these polynomials. We shall adopt in this paper a notation used by Al-Salam [2]. In the notation of hypergeometric series, the generalized Bessel polynomials due to Chatterjea [7] are given by h xi (1.3) yn(α) (x; k) = 2 F0 −n, kn + α; −; − , 2 where k is a positive integer and n = 0, 1, 2, . . . . Sometimes we shall find it convenient to consider the following polynomial θn(α) = xn yn(α) (x; k).

(1.4)

2. Recurrence relations From (1.3) it is easy to find that yn(α+1) (x; k) − yn(α) (x; k) =

(2.1)

nx (α+k+1) (x; k). y 2 n−1

This suggests the difference formula ∆α yn(α) (x; k) =

(2.2)

nx (α+k+1) y (x; k) 2 n−1

r where ∆α f (α) = f (α + 1) − f (α) and ∆r+1 α f (α) = ∆α ∆α f (α). In particular x n (2.3) ∆nα yn(α) (x; k) = . 2 Now Newton’s formula X u f (α + u) = ∆r f (α) r r

and (2.3) imply (2.4)

yn(α+u) (x; k)

=

X u r

or equivalently (2.5)

θn(α+u) (x; k)

x r n! (α+r+rk) yn−r (x; k) r (n − r)! 2

X u n!2−r (α+r+rk) (x; k). = θ r (n − r)! n−r r

Also, from (1.3) we find that (2.6)

d (α) 1 (α+r+rk) yn (x; k) = n(kn + α + 1)yn−1 (x; k). dx 2

GENERALIZED BESSEL POLYNOMIAL

95

From (2.6) and (2.2), we see that the polynomial given in (1.3) satisfy the mixed equation x d (α) y (x; k). kn + α + 1 dx n The following recurrence relation can easily be verified: ∆α yn(α) (x; k) =

(2.7)

(α−k+1)

(2.8)

yn+1

(x; k) − yn(α) (x; k) =

1 x(kn + n + α + 2)yn(α+1) (x; k). 2

3. Integral representations It is easy to derive the following integral representations for the Chatterjea’s generalized Bessel polynomials (1.3): Z1

tβ−1 (1 − t)γ−1 yn(α) (xt; k) =

Z1

tβ−1 (1 − t)γ−1 yn(α) (x(1 − t); k) =

0

(3.1)

0

(3.2)

Z∞

(3.3)

  −n, β, kn + α + 1; Γ(β)Γ(γ) − x2  . = 3 F1  Γ(β + γ) β + γ;

st kn+α

e t

0

Z1

(α) ym

0

=−

(3.4)

  −n, γ, kn + α + 1; Γ(β)Γ(γ)  − x2  . = 3 F1 Γ(β + γ) β + γ; sxt 1+ 2

n

dt =

Γ(kn + α + 1) (α) yn (x; k). skn+α+1

t t (β) ; k yn ; k x−km−α−1 (1 − x)−kn−β−1 dx x 1−x

π sin π(α + β)Γ(km + kn + α + β + 1) (α+β) y (t; k). sin πα sin πβΓ(km + α + 1)Γ(kn + β + 1) m+n

Some interesting particular cases of (3.1), (3.2), (3.3) and (3.4) are as follows: (i) Taking β = kn + 1, γ = α in (3.1), we get (3.5)

Z1 0

tkn (1 − t)α−1 yn(α) (xt; k)dt =

Γ(kn + 1)Γ(α) (0) y (x; k). Γ(kn + α + 1) n

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(ii) Taking β = n + α + 1, γ = kn − n in (3.1), we obtain (3.6)

Z1 0

Γ(kn − n)Γ(n + α + 1) (α) yn (x; k). Γ(kn + α + 1)

tn+α (1 − t)kn−n−1 yn(α) (xt; k)dt =

(iii) Replacing β by kn + α + β + 1 and taking γ = 1 − β in (3.1), we get Z1

(3.7)

0

tkn+α+β (1 − t)β yn(α) (xt; k)dt = =−

πΓ(kn + α + β + 1) y (α+β) (x; k). sin πβΓ(1 + β)Γ(kn + α + 1) n

(iv) Replacing β by kn + α − β + 1 and taking γ = 1 + β in (3.1), we have (3.8)

Z1 0

tkn+α−β (1 − t)β yn(α) (xt; k)dt =

Γ(kn + α − β + 1)Γ(β) (α−β) yn (x; k). Γ(kn + α + 1)

Results similar to (3.5), (3.6), (3.7) and (3.8) will hold by suitable selection of β and γ in (3.2) also. (v) Taking γ = 1 in (3.3), it becomes n Z∞ xt dt = Γ(kn + α + 1)yn(α) (x; k). e−t tkn+α 1 + (3.9) 2 0

(vi) Taking γ = 1 in (3.1), it reduces to   Z1 −n, β, kn + α + 1; 1 − x2  . tβ−1 yn(α) (xt; k)dt = 3 F1  (3.10) β β + 1; 0 (vii) Taking β = 1 in (3.2), it reduces to

(3.11)

Z1 0

(1 − t)γ−1 yn(α) (x(1 − t); k)dt =



1  3 F1 γ

−n, γ, kn + α + 1; γ + 1;

(viii) Taking β = kn + α, γ = 1 in (3.1), it reduces to Z1

(3.12)

tkn+α−1 yn(α) (xt; k)dt =

− x2  .

1 y (α+1) (x; k). kn + α n

0

(ix) Taking β = 1 and γ = kn + α in (3.2), it becomes (3.13)

Z1 0

(1 − t)kn+α−1 yn(α) (x(1 − t); k)dt =



1 y (α+1) (x; k). kn + α n


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(x) For n = 0, (3.4) becomes Z1 t (α) ; k x−km−α−1 (1 − x)−β−1 dx ym x 0

=−

(3.14)

π sin π(α + β)Γ(km + α + β + 1) (α+β) (t; k). y sin πα sin πβΓ(km + α + 1)Γ(1 + β) m

(xi) For m = 0, (3.4) becomes Z1 t (β) yn ; k x−α−1 (1 − x)−kn−β−1 dx 1−x 0

(3.15)

=−

π sin π(α + β)Γ(kn + α + β + 1) (α+β) (t; k). y sin πα sin πβΓ(α + 1)Γ(kn + β + 1) n

We now give a two dimensional version of (3.4). It is given by ZZ t t t (α) ym ; k yn(β) ; k yp(γ) ;k u v 1−u−v

u+v≤1

u−km−α−1 v −kn−β−1 (1 − u − v)−kp−γ−1 dudv π 2 sin π(α + β + γ) = sin πα sin πβ sin πγ Γ(km + kn + kp + α + β + γ + 1) (α+β+γ) (3.16) y (t; k) Γ(km + α + 1)Γ(kn + β + 1)Γ(kp + γ + 1) m+n+p where the integration is over the interior of the triangle bounded by the u and v axes and the line u + v = 1. The extension of (3.16) to higher dimensions is immediate. 4. Some characterizations In this section we obtain some characterizations of Chatterjea’s generalized Bessel polynomials (1.3) similar to those obtained by Al-Salam [2] for Bessel polynomial. (α)

Theorem 4.1. Given a sequence {fn (x; k)} of polynomials in x where (α) deg fn (x; k) = n, and α is a parameter, such that d (α) 1 (α+k+1) (4.1) f (x; k) = n(kn + α + 1)fn−1 (x; k) dx n 2 (α)

(α)

(α)

and fn (0; k) = 1. Then fn (x; k) = yn (x; k). Proof. Let fn(α) (x; k) =

x r . Cr (α, n, k) − 2 r=0

n X

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Then by (3.1), we have Cr (α, n, k) = −

n(kn + α + a) Cr−1 (α + k + 1, n − 1, k). r

Since C0 (α, n, k) = 1, (−n)r (kn + α + 1)r , r!

Cr (α, n, k) = which proves the theorem.

Another characterization is suggested by (2.2) as given in the following theorem: (α)

Theorem 4.2. Given a sequence of functions {fn (x; k)} such that (4.2)

∆α fn(α) (x; k) =

(4.3)

fn(α) (0; k) = 1, (α)

1 (α+k+1) nxfn−1 (x; k) 2 (α)

f0 (x; k) = 1.

(α)

Then fn (x; k) = yn (x; k). (α)

Proof. We observe from (4.2) that fn (x; k) is a polynomial in α of degree n. Hence we can write fn(α) (x; k) =

n X

Cr (n, x)

r=0

Hence (4.2) gives

(kn + α + 1)r . r!

nx Cr−1 (n − 1, x). 2 From this recurrence and condition (4.3), we obtain x r . Cr (n, x) = (−n)r − 2 This proves the theorem. Cr (n, x) =

Now equation (2.7) gives the following: (α)

(α)

Theorem 4.3. Let the sequence {fn (x; k)}, where fn (x; k) is a polynomial of degree n in x, and α is a parameter, satisfy ∆α fn(α) (x; k) =

(4.4) such that (4.5) (α)

x d (α) f (x; k) kn + α + 1 dx n

h xi fn(0) (x; k) = 2 F0 −n, kn + 1; −; − . 2 (α)

Then fn (x; k) = yn (x; k).


99

The proof is similar to that of Theorem 4.1 and 4.2. (α) Similarly (2.8) suggests yet another characterization of yn (x; k) given in the form of the following theorem: (α)

Theorem 4.4. Given a sequence of functions fn (x; k)} such that 1 (α−k+1) (4.6) fn+1 (x; k) − fn(α) (x; k) = x(kn + n + α + 2)fn(α+1) (x; k) 2 and (α) f0 (x; k) = 1 for all x and α. (α)

(α)

Then fn (x; k) = yn (x; k). The proof of this theorem follows by induction on n. 5. Rodrigue’s formula Krall and Frink [10] gave the following Rodrigue’s type formula for the Bessel polynomials yn (x, a, b): dn (5.1) yn (x, a, b) = b−n x2−a eb/x n x2n+a−2 e−b/x . dx It is not difficult to establish the following Rodrigue’s type formula for the (α) polynomial yn (x; k): h i 1 2 d 2 (5.2) yn(α) (x; k) = n kn−n+α e x Dn xkn+n+α e− x , D ≡ . 2 x dx References

[1] R. P. Agarwal, On Bessel Polynomials, Canad. J. Math. 6 (1954), 410–415. [2] W. A. Al-Salam, The Bessel Polynomials, Duke Math. J. 24 (1957), 529–545. [3] F. Brafman, A set of generating functions for Bessel polynomials, Proc. Am. Math. Soc. 4 (1953), 275–277. [4] J. L. Burchnall, The Bessel polynomials, Canad. J. Math. 3 (1951), 62–68. [5] L. Carlitz, On the Bessel polynomials, Duke Math. J. 24 (1957), 151–162. [6] L. Carlitz, A characterization of the Laguerre polynomials, Monatshefte f¨ ur Mathematik, 66 (1962), 389–392. [7] S. K. Chatterjea, Some generating functions, Duke Math. J. 32 (1965), 563–564. [8] D. Dickinson, On Lommel and Bessel polynomials, Proc. Am. Math. Soc. 5 (1954), 946–956. [9] E. Grosswald, On some algebraic properties of the Bessel polynomials, Trans. Am. Math. Soc. 71 (1951). [10] H. L. Krall and O. Frink, A new class of orthogonal polynomials the Bessel polynomials, Trans. Am. Math. Soc. 65 (1949), 100–115. [11] E. D. Rainville, Generating functions for Bessel and related polynomials, Canad. J. of Math. 5 (1953), 104–106. [12] E. D. Rainville, Special Functions, Macmillan, New York, Reprinted by Chelsea Publ. Co., Bronx, New York, 1971. [13] H. M. Srivastava and H. L. Manocha, A treatise on generating functions, John Wiley and Sons (Halsted Press), New York, Ellis Horwood, Chichester, 1984.

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[14] L. Toschano, Osservazioni, confronti e complenti su particolari polinomi ipergeometrici, Le Matematiche, 10 (1955), 121–l33.

Department of Applied Mathematics, Faculty of Engineering, A. M. U., Aligarh-202002, India Received : 28.09.1998.