JACOBI TYPE AND GEGENBAUER TYPE GENERALIZATION OF CERTAIN POLYNOMIALS. Mumtaz Ahmad Khan and Mohammad Asif. 1. Introduction

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1 MATEMATIQKI VESNIK 64 (0) June 0 originalni nauqni rad research paper JACOBI TYPE AND GEGENBAUER TYPE GENERALIZATION OF CERTAIN POLYNOMIALS Mumtaz Ahmad Khan and Mohammad Asif Abstract. This paper deals with the Jacobi type and Gegenbauer type generalizations of certain polynomials and their generating functions. Relationships among those generalized polynomials have also been indicated.. Introduction In 947 Sister Celine (Fasenmyer ) obtained some basic formal properties of the hypergeometric polynomials. Sister Celine s polynomials are defined by the following generating relation which yields ( t) rf s a... a r ; b... b s ; f n a... a p ; b... b s ; x t n (.) f(a i ; b i ; x) f n (a a a r ; b b b s ; x) r+ F s+ n n + a... a r ; b... b s ; x (n is non negative integer) in an attempt to unify and to extend the study of certain sets of polynomials which have attracted considerable attention. Her polynomials include as special cases Legendre s polynomials P n ( x) some special Jacobi polynomials Rice s polynomials H n (ξ p ν) Bateman s Z n (x) F n (z) and Pasternak s F m n (z) which is a generalization of Bateman s polynomials F n (z). 00 AMS Subject Classification: 33C45. Keywords and phrases: Jacobi type and Gegenbauer type generalization of Sister Celine s polynomials Bateman s polynomials Pasternack s polynomials Hahn polynomial Rice polynomials generating functions. 47

2 48 M. A. Khan M. Asif.. Bateman s polynomials Bateman defined the following polynomials (see 8 7; eq. () p. 89) n n + F n (z) 3 F ( + z); ; Bateman 7 8 obtained the following generating functions + z; ; F n (z)t n t F F n (z ) F n (z)t n.. Pasternack s polynomials t ( t) 3 F 3 + z; ; The generalization of Bateman s polynomial due to Pasternack is given below Fn m n n + (z) 3 F (z + m + ); m + ; which is a generalization of Bateman s polnomials F n (z). Generating function of generalization of Pasternack s polynomials is given below Fn m (z)t n ( t) F (z + m + ); m + ; In 936 Bateman 7 was interested in constructing inverse Laplace transforms. For this purpose he introduced the following polynomial n n + ; Z n (x) F x ; By theorem 48 (see 7 p. 37) Rainville writes the generating function as follows Z n (x)t n t F n + ; ; In 939 Pasternack 7 obtained the following generating function of Bateman s polynomials F m (n ) (t)n e t Z m (t) ( + t F n (m )t n ( t) m ( + t) m ) P m t.3. Rice s polynomials S. O. Rice made a considerable study of the polynomials defined by n n + ξ; H n (ξ p ν) 3 F ν p;

3 Jacobi type and Gegenbauer type generalization The generalized Rice s Polynomial due to Khandekar 4 is given below H (αβ) n n + α + β + ξ; n (ξ p ν) 3 F ν ( + α) n + α p; (.) Generating function of the generalized Rice polynomial due to Khandekar 4 is given below (α + ) n H (αβ) (; α + β + ) ξ; n (ξ p ν)t n αβ 3F ( + α) n + α p;.4. Hahn polynomials Hahn polynomial is defined as Q n (x; α β N) 3 F n n + α + β x; + α N; 4νt α β > n x 0... N. The following generating functions are satisfied by the Hahn polynomial (N) n x N; (β + ) n Q n(x; α β N)t n F x; α + ; t F β + ; x 0... N and (N) n x x + β + N + ; (β + ) n Q n(x; α β N)t n F 0 ; F 0 x N x + α + ; ; t t t x 0... N. Motivated by the Jacobi type generalization of the Rice s polynomials obtained by Khandekar 4 we aim here to obtain Jacobi type generalization of the polynomials mentioned in the first section of this paper.. Jacobi type generalization of certain polynomials and their generating functions Before obtaining the Gegenbauer type generalization of polynomials we shall first discuss the Jacobi type generalization of polynomials. The Gegenbauer type generalizations shall be discussed in the next section of this paper. The Jacobi type generalization of Sister Celine s polynomial is given below... Jacobi type generalization of Sister Celine s polynomial Substituting λ c p r q s r s and µ α+ in the Eq. (3.6) of the reference 3 the following generalized Sister Celine s polynomials are obtained by means of the following generating relation ( t) c c +rf +c a... a r ; +s + α + α b... b s ; c f +c a... a p ; n + α + α b... b s ; x t n

4 50 M. A. Khan M. Asif which produces the following relation c f +c a... a p ; n + α + α b x... b s ; (c) n Case (i). For α 0 it reduces to c ( t) c +r F +c a... a r ; +s b... b s ; +rf +s n n + c a... a r ; + α + α b... b s ; c f +c a... a r ; n b... b s ; x x t n Case (ii). For α 0 and c it reduces to original Sister Celine polynomial (see 7; eq. () pp. 90 of Rainville) and eq. (.). Case (iii). For c + α + β it gives Jacobi type generalization of Sister Celine s polynomial ( + α + β) n n n + + α + β a... a r ; +rf +s + α + α b x t... b s ; n +α+β +α+β ( t) +α+β +r F a... a r ; +s + α + α b... b s ; For α β we get the following ultraspherical type generalization of Sister Celine s polynomials ( + α) n x t n +rf +s n n + α + a... a r ; + α + α b... b s ; ( t) +α r F s Case (iv). For c + α + β and r s 0 we have ( + α + β) n n n + + α + β; F + α + α; x t n ( t) +α+β F a... a r ; b... b s ; +α+β +α+β ; + α + α; Case (v). Whenα β r s 0 we have the following form of Sister Celine s polynomial n n + ; F ; x t n t exp ( t)... Jacobi type generalization of Bateman s polynomials In 999 M. A. Khan and A. K. Shukla 3 defined the Jacobi type generalization of Bateman s polynomials as follows F n (αβ) n n + α + β + (p z) 3 F ( + z);. (.) + α p;

5 Jacobi type and Gegenbauer type generalization... 5 We define the generating function for the Jacobi type generalization of Bateman s polynomials defined by Khan and Shukla. ( + n) α+β F n (αβ) (p z)t n () α+β ( + n) α+β () α+β ( + n) α+β () α+β n k0 n k0 n k0 (n) k (n + α + β + ) k ( ( + z)) kt n () k () n (n k)! () k () n+α+β+k (n k)! ( + α) k (p) k k! (n + α + β + ) k ( ( + z)) kt n ( + α) k (p) k k! ( ( + z)) kt n () α+β ( + α) k (p) k k! () k () n+α+β+k ( ( + z)) kt n+k k0 () α+β ( + α) k (p) k k! () α+β ( + α + β) k ( ( + z)) k(t) k ( + α + β + k) n t n k0 () α+β ( + α) k (p) k k! k ( +α+β ) k ( +α+β ) k ( ( + z)) k(t) k ( t) +α+β k0 ( + α) k (p) k k! ( t) k +α+β +α+β ( t) +α+β 3 F ( + z); + α p;. Another kind of generating function can also be defined for the Jacobi type generalization of the Bateman s polynomials given below ( + n) α+β F n (αβ) (p z ) F n (αβ) (p z)t n () α+β ( + n) α+β n (n) k (n + α + β + ) k () α+β k0 ( + α) k (p) k k! ( + n) α+β n () k () n (n k)! n () α+β n n k n k0 n n k0 k0 k () k+ () n+α+β+k (n k)! () k+ () n+α+β+k+ (n k )! k0 () k () n+α+β+k (n )! () k () n+α+β+k+ (n + α + β + ) k ( + α) k (p) k k! {( ( + z)) k ( ( + z)) } k t n { (k) ( } ( + z)) t n k ( ( + z)) kt n () α+β ( + α) k (p) k (k )! ( ( + z)) kt n () α+β ( + α) k+ (p) k+ k! ( ( + z)) kt n+k () α+β ( + α) k+ (p) k+ k! ( ( + z)) kt n+k () α+β ( + α) k+ (p) k+ k! () α+β ( + α + β) n+k+ ( ( + z)) k(t) k t n () α+β ( + α) k+ (p) k+ k!

6 5 M. A. Khan M. Asif ( + α + β) (3 + α + β) n+k ( ( + z)) k(t) k t n k0 ( + α) k+ (p) k+ k! (3 + α + β) k ( ( + α + β) ( + z)) k(t) k (3 + α + β + k)t n k0 ( + α) k+ (p) k+ k! ( + α + β) k ( 3+α+β ) k ( 4+α+β ) k ( ( + z)) k(t) k ( t) 3+α+β ( + α)( + α) k (p)(p + ) k k! ( t) k k0 ( + α + β) ( t) 3+α+β ( + α)(p) 3 F 3+α+β 4+α+β ( + z); + α p + ;. For α β we get ultraspherical type generalization of Bateman s polynomials F n (αα) n n + α + (p z) 3 F ( + z); + α p; Generating function for these polynomials is given below ( + n) α {F n (αα) () α (p z ) F n (αα) (p z)}t n ( + α) 3+α ( t) 3+α ( + α)p F ( + z); p + ;.3. Jacobi type generalization of Pasternack s polynomial By substituting z z + m p + m in eq. (.) Khan and Shukla 3 obtained Jacobi type generalization of Pasternack s generalized Bateman s polynomial Fn m (z) given below F nm (αβ) n n + α + β + (z) 3 F ( + z + m);. (.) + α + m; We derive here the generating function of these polynomials ( + n) α+β F nm (αβ) (z)t n () α+β +α+β +α+β ( t) +α+β 3 F ( + z + m); + α + m;. Substitution of α β gives us its ultraspherical type generalization F nm (αα) n n + α + (z) 3 F ( + z + m);. (.3) + α + m; The following is the generating function of the ultraspherical type generalized of the polynomial (.3). ( + n) α F nm (αα) (z)t n +α () α ( t) +α F ( + z + m); + m;

7 Jacobi type and Gegenbauer type generalization Jacobi type generalization of Bateman s polynomial Z n (x) Jacobi type generalization of Bateman s polynomials Z n (x) was considered to be new in the last decade. For α β ν it reduces to Gegenbauer generalization of the Bateman s polynomials and also for α β 0 it reduces to Bateman s polynomial. Khan and Shukla 3 adopted the symbol Z n (αβ) (b x) to denote the Jacobi type generalization of Bateman s polynomial Z n (x). Z n (αβ) n n + α + β + ; (b x) F x. (.4) + α b + ; We determine the generating function for the polynomial Z n (αβ) (b x) ( + n) α+β Z n (αβ) (b x)t n ( + n) α+β () α+β k0 k0 ( + n) α+β n k0 () α+β n k0 () α+β n k0 () k () n (n k)! () k () n+α+β+k (n k)! () k () n+α+β+k () α+β ( + α + β) k (xt) k () α+β ( + α) k (b + ) k k! ( t) +α+β k0 (n) k (n + α + β + ) k x k t n ( + α) k (b + ) k k! (n+α+β+) k x k t n (+α) k (b+) k k! x k t n () α+β (+α) k (b+) k k! x k t n+k () α+β (+α) k (b+) k k! ( + α + β + k) n t n k ( +α+β ) k ( +α+β ) k (xt) k ( + α) k (b + ) k k! ( t) k ( t) +α+β F +α+β +α+β ; + α b + ; The generating function given below establishes a relation between F mn (αβ) (m t); we have Z (αβ) n m m + α + β + mn (z) 3 F ( + z + n); + α + m; F (αβ). and (.5) F mn (αβ) (n ) (t)n n k0 (m) k (m + α + β + ) k (n) k (t) n ( + α) k ( + m) k k! n (m) k (m + α + β + ) k () k () n (t) n k0 ( + α) k ( + m) k k! (n k)! (m) k (m + α + β + ) k () k (t) n+k k0 ( + α) k ( + m) k k! (t) n (m) k (m + α + β + ) k t k ( + α) k ( + m) k k! k0

8 54 M. A. Khan M. Asif e t m m + α + β + ; F t e t Z m (αβ) (m t). + α + m; For α βwe get the ultraspherical type generalization of it as given below: F mn (αα) m m + α + (z) 3 F ( + z + n); + α + m; F mn (αα) (n ) (t)n m m + α + ; e t F t e + α + m; t Z m (αα) (m t). In particular for α β we obtain the ultraspherical generalization of Bateman s polynomial due to Khan and Shukla 3 Z n (αα) n n + α + ; (b x) F x + α b + ; Generating function for the ultraspherical generalized Bateman s polynomials is given below ( + n) α () α Z (αα) n (b x)t n +α ( t) +α F ; b + ; Substitution of α β reduces Khandekar s polynomial to ultraspherical type generalization of Rice s polynomials therefore we have H (αα) n n + α + ξ; n (ξ p ν) 3 F ν ( + α) n + α p; Generating functions for the ultraspherical type generalized Rice s polynomial is given below (α + ) n H n (αα) (ξ p ν)t n ( + α) n (α + ) n (+α) n n (n) k (n + α + ) k (ξ) k ν k t n ( + α) n k0 ( + α) k (p) k k! n () k (α + ) n (n + α + ) k (ξ) k ν k t n (n k)!( + α) k (p) k k! k0 k0 k0 n k0 k0 () k (α + ) n+k (ξ) k ν k t n (n k)!( + α) k (p) k k! (α + ) n+k (ξ) k (νt) k t n ( + α) k (p) k k! (α + + k) n t n (α + ) k (ξ) k (νt) k ( + α) k (p) k k! k ( + α) k( + α) k (ξ) k (νt) k ( + α) k (p) k k! ( t) α++k ( + α) k(ξ) k (4νt) k ( t) α+ (p) k k!( t) k k0 ( t) α+ F + α ξ; p; 4νt

9 Jacobi type and Gegenbauer type generalization Ultraspherical type generalization of the Hahn polynomial is given below n n + α + x; Q n (x; α α N) 3 F (.6) + α N; Polynomials given by (.6) satisfy the following generating functions ( + α) n Q n (x; α α N)t n ( t) α+ F + α x; N;. 3. Gegenbauer type generalization of certain polynomials and their generating functions In this section we determine Gegenbauer type generalization of the polynomials mentioned in the introduction section of this paper. 3.. Gegenbauer type generalization of the Sister Celine s polynomial Gegenbauer type generalization of the Sister Celine s polynomials is given below f n (ν) (a a r ; b b s ; x) (ν) n (ν) n n n + ν a a r ; +rf +s + ν ν b b s ; +rf +s n n + ν a a r ; + ν ν b b s ; Generating function of the Gegenbauer type generalization of the Sister Celine s polynomial is given below x t n (ν) n n (n) k (n + ν) k (a ) k (a r ) k x k t n k0 ( + ν) k(ν) k (b ) k (b s ) k k! n () k (ν) n+k (a ) k (a r ) k x k t n k0 (n k)!( + ν) k(ν) k (b ) k (b s ) k k! () k (ν) n+k (a ) k (a r ) k x k t n+k k0 k!( + ν) k(ν) k (b ) k (b s ) k (ν) k (a ) k (a r ) k (xt) k (ν + k) n t n k0 k!( + ν) k(ν) k (b ) k (b s ) k k (a ) k (a r ) k (xt) k (ν + k) n t n k0 k!(b ) k (b s ) k (a ) k (a r ) k () k ( t) ν k!(b ) k (b s ) k ( t) k k0 ( t) ν r F s a a r ; b b s ; x

10 56 M. A. Khan M. Asif 3.. Gegenbauer type generalztion of the Bateman s polynomial By substituting α β ν the Gegenbauer type generalization of the Bateman s polynomial F n (z) is obtained as given below n n + ν F n (ν) (p z) 3 F ( + z); ν + p;. (3.) One of the generating functions of polynomials (3.) is determined below (ν) n F n (ν) (p z)t n ν ( t) ν F ( + z); p;. Another generating function of polynomial (3.) is obtained below (ν) n {F n (ν) (p z ) F n (ν) (p z)}t n (ν) ν + ( t) +ν (ν + )(p) 3 F ν + ( + z); ν + 3 p + ; By setting α β ν in the Eq. (.4) another Gegenbauer type generalization of Bateman s polynomials Z n (ν) (b x) is given below Z n (ν) n n + ν; (b x) F ν + + b; x (3.) Gegenbauer type generalized Bateman s polynomial (3.) satisfies the following generating function (ν) n Z n (ν) (b x)t n ν; ( t) ν F + b; Gegenbauer type generalization of Pasternack s polynomial By setting α β ν and + m p in Eq. (.) we define here the Gegenbauer type generalization of Pasternack s generalization of the Bateman s polynomial Fn m (p z) as given below n n + ν F nm(p (ν) z) 3 F (z + m + ); ν + p;. (3.3) For the above polynomial (3.3) we easily obtain the following generating function relation (ν) n F nm(p (ν) z)t n ( t) ν F ν (z + m + ); p; Now substituting α β ν in Eq. (.5) the following polynomial is obtained m m + ν F mn(z) (ν) 3 F ( + z + n); ν + + m; whose generating function is determined as given below F mn(n (ν) ) (t)n m m + ν; e t F ν + + m; t e t Z m (ν) (m t).

11 Jacobi type and Gegenbauer type generalization Gegenbauer type Generalization of Rice s polynomials With the steps taken by Khandekar 4 in the definition of Jacobi type generalization of Rice s polynomials by putting α β ν in Eq. (.) we define Gegenbauer generaliation of Rice s polynomials ν ( ) ν + n H (ν) n (ξ p ν) 3 F n n + ν ξ; ν + p; We define the generating functions of the Gegenbauer type generalization of the Rice s polynomial as follows (ν) n (ν + ) H n (ν) (ξ p ν)t n ν ξ; n ( t) ν F p; 4νt 3.5. Gegenbauer type generalization of Hahn polynomials We define the Gegenbauer type generalization of Hahn s polynomial as given below Q n (ν) n n + ν x; (x; N) 3 F + ν N;. The following generating function is satisfied by the Hahn s polynomial (ν) n Q (ν) n (x; N)t n (ν) n n (n) k (n + ν) k (x) k t n k0 ( + ν) k(n) k k! n () k (ν) n+k (x) k t n k0 (n k)!( + ν) k(n) k k! () k (ν) n+k (x) k t n+k k0 ( + ν) k(n) k k! () k (ν) k (x) k t k (ν + k) n t n k0 ( + ν) k(n) k k! k (ν) k ( + ν) k(x) k (t) k ( t) ν k0 ( + ν) k(n) k k!( t) k ν x; N; ( t) ν F. Acknowledgements. The second author wishes to express his heartfelt thanks to the Human Resource Development Group Council of Scientific & Industrial Research of India for awarding Senior Research Fellowship (NET)(F.No. 0-(5)/ 005(i)- E.U.II). REFERENCES T.S. Chihara An Introduction to Orthogonal Polynomials Gordon and Breach New York London and Paris 978. Fasenmyer Sister M. Celine Some generalized hypergeometric polynomials Bull. Amer. Math. Soc. 53 (947) M.A. Khan A.K. Shukla On some generalized Sister Celine s polynomials Czech. Math. J. 49 (999)

12 58 M. A. Khan M. Asif 4 P.R. Khandekar On a generalization of Rice s polynomial I. Proc. Nat. Acad. Sci. India Sect. A34 (964) J.D.E. Konhauser Some properties of biorthogonal polynomials J. Math. Anal. Appl. (965) J.D.E. Konhauser Biorthogonal polynomials suggested by Laguerre polynomials Pacific J. Math. (967) E.D. Rainville Special Functions Macmillan New York; Reprinted by Chelse Publ. Co. Bronx New York H.M. Srivastava H.L. Manocha A Treatise on Generating Functions John Wiley and Sons (Halsted Press) New York; Ellis Horwood Limited Chichester 984. (received ; in revised form 0.0.0; available online ) Department of Applied Mathematics Faculty of Engineering Aligarh Muslim University Aligarh- 000 U.P. INDIA mumtaz ahmad khan 008@yahoo.com mohdasiff@gmail.com