Proyecciones Journal of Mathematics Vol. 29, N o 3, pp. 201-207, December 2010. Universidad Católica del Norte Antofagasta - Chile POLYNOMIAL SETS GENERATED BY e t φ(xt)ψ(yt) MUMTAZ AHMAD KHAN ALIGARH MUSLIM UNIVERSITY, INDIA and BAHMAN ALIDAD GOLESTAN UNIVERSITY, IRAN Received : August 2009. Accepted : October 2010 Abstract The present paper deals with two variables polynomial sets generated by functions of the form e t φ(xt)ψ(yt). Itsspecialcaseanalogous to Laguerre polynomials have been discussed.
202 Mumtaz Ahmad Khan and Bahman Alidad 1. INTRODUCTION Laguerre polynomials L n (α) (x) possessthegeneratingrelation (see Rainville [8], pp. 130) e t 0F 1 ( ; 1+α; xt) By studying generating relation L n (α) (x)t n (1.1) (1 + α) n e t ψ(xt) σ n (x) t n (1.2) One arrives at properties held by L (α) n (x) (see Rainville [8], p. 132-133) Motivated by (1.2) an attempt has been made to study two variable polynomials similar to one given in (1.2) and generated by functions of the form e t φ(xt)ψ(yt). 2. TWO VARIABLE POLYNOMIAL SETS ANALOGOUS TO (1.2) Let us consider the generating relation of the type Let Then e t φ(xt) ψ(yt) σ n (x, y) t n (2.1) F e t φ(xt) ψ(yt) (2.2) F x tet φ 0 ψ (2.3) F y tet φψ 0 (2.4) F t et φψ+ xe t φ 0 ψ + ye t φψ 0 (2.5) Eliminating φ, φ 0, ψ and ψ 0 from the four equations (2.2), (2.3), (2.4) and (2.5), we obtain x + F t F t F (2.6) y t
Polynomial sets generated by e t φ(xt)ψ(yt) 203 Since Equation (2.6) yields F e t φ(xt) ψ(yt) σ n (x, y) t n x + y σ n (x, y)t n y nσ n (x, y) t n σ n (x, y)t n+1 from which the next theorem follows. Theorem 1 : From e t φ(xt) ψ(yt) σ n (x, y) t n σ n 1 (x, y)t n n1 it follows that σ 0(x, y) y σ 0(x, y) 0andforn 1, x + y σ n (x, y) nσ n (x, y) σ n 1 (x, y) (2.7) y Next, let us assume that the functions φ and ψ in(2.1)havetheformal power - series expansion and Then (2.1) yields φ(u) γ n u n (2.8) ψ(v) δ n v n (2.9) Ã σ n (x, y) t n X t n!ã!ã X! γ n x n t n X δ n y n t n
204 Mumtaz Ahmad Khan and Bahman Alidad so that n r X r0 s0 γ r δ s x r y s t n (n r s)! Now consider the sum σ n (x, y) n r X r0 s0 γ r δ s x r y s (n r s)! (2.10) n r (c) n σ n (x, y) t n X (c) n γ r δ s x r y s t n (n r s)! r0 s0 (c) n+r+s γ r δ s x r y s t n+r+s r0 s0 (c) r+s γ r δ s (xt) r (yt) s X (c + r + s) n t n r0 s0 (c) r+s γ r δ s (xt) r (yt) s (1 + t) r0 s0 c+r+s (2.10) We thus arrive at the following theorem: Theorem 2 : From e t P φ(xt) ψ(yt) σ n (x, y) t n P, φ(u) follows that for arbitrary c in which (1 t) c F xt 1 t, yt 1 t γ n u n P, ψ(v) δ n v n it (c) n σ n (x, y) t n (2.11) F (u, v) (c) n+k γ n δ k u n v k (2.12) k0 The role of Theorem 2 is as follows: If a set σ n (x, y) hasagenerating function of the form e t φ(xt) ψ(yt), Theorem 2 yields for σ n (x, y) another generating function of the form exhibited in (2.11). For instance, if φ(u)and ψ(v) arespecified p F q, the theorem gives for σ n (x, y) aclass(c arbitrary) of generating functions involving two variables hypergeometric functions.
Polynomial sets generated by e t φ(xt)ψ(yt) 205 Let us now apply Theorems 1 and 2 to Laguerre polynomials of two variables L n (α,β) (x, y) due to S. F. Ragab [7] defined by L (α,β) Γ(n + α +1)Γ(n + β +1) n (x, y) r0 ( y) r L α n r(x) r! Γ(α + n r +1)Γ(β + r +1) (2.13) Where L n (α) (x) is the well - known Laguerre polynomials of one variable. The definition (2.13) is equivalent to the following explicit representation of L (α,β) n (x, y), given by Ragab: L (α,β) n (x, y) (α +1) n(β +1) n () 2 n r X r0 s0 ( n) r+s x s y r (α +1) s (β +1) r r! s! (2.14) Later, the same year Chatterjea [1] gave the following generating function for L n (α,β) (x, y): e t L n (α,β) (x, y) t n 0F 1 ( ; α +1; xt) 0 F 1 ( ; β +1; yt) (α +1) n (β +1) n (2.15) We use Theorem 1 to conclude that L (α,β) 0 (x, y) is a constant and, and for n 1. x + y y L n (α,β) (x, y) nl n (α,β) (x, y) (α + n)(β + n) n L (α,β) n 1 (x, y) (2.16) In applying theorem 2 to Laguerre polynomials of two variables L n (α,β) (x, y), L(α,β) n (x,y) note that σ n (x, y) and that Then γ n (α+1) n (β+1) n φ(u) 0 F 1 ( ; 1+α; u) ψ(v) 0 F 1 ( ; 1+β; v) ( 1)n (1+α) n, δ n ( 1)n (1+β) n and ( 1) n u n (1+α) n ( 1) n v n (1+β) n
206 Mumtaz Ahmad Khan and Bahman Alidad F (u, v) (c) n+k γ n δ k u n v n (c) n+k ( 1) n+k u n v k k! (1+α) k0 k0 n (1 + β) k Therefore Theorem 2, yields ψ 2 [c;1+α, 1+β; u, v] (1 t) c ψ 2 c;1+α, 1+β; xt 1 t, yt 1 t (c) n L (α,β) n (x, y)t n (1 + α) n (1 + β) n (2.17) a class of generating relations for L n (α,β) (x, y) due to M.A. Khan and A.K. Shukla [2]. Concluding Remark Application of the theorems given in this paper have already been shown in case of Laguerre polynomials of two variables. Thus, this class of product may be used whenever Laguerre polynomials of two variables occur. References [1] CHATTERJEA, S. K. A note on Laguerre polynomials of two variables, Bull. Cal. Math. Soc., Vol. 83, pp. 263, (1991). [2] KHAN, M. A. AND SHUKLA, A. K. On Laguerre polynomials of several variables, Bull. Cal. Math. Soc., Vol. 89, pp. 155-164, (1997). [3] KHAN, M. A. AND SHUKLA, A. K. AnoteonLaguerrepolynomials of m variables, Bulletin of the Greek Mathematical Society., Vol. 40, pp. 113-117, (1998). [4] KHAN, M. A. AND ABUKHAMMASH, G. S. On Hermite polynomials of two variables suggested by S. F. Ragab s Laguerre polynomials of two variables, Bull. Cal. Math. Soc., Vol. 90, pp. 61-76, (1998).
Polynomial sets generated by e t φ(xt)ψ(yt) 207 [5] KHAN, M. A. AND AHMAD, K. On a general class of polynomials L n (α,β;γ,δ) (x, y) of two variables suggested by the polynomials L n (α,β) (x, y) of Ragab and L n (α,β) (x) of Prabhakar and Rekha, Pro Mathematica, Vol. xix/nos. 37-38, pp. 21-38, (2005). [6] KHAN, M. A. AND ALIDAD, B. Polynomial sets generated by functions of the form G(2xt t 2 )K(2yt t 2 ),Communicated for publication. [7] RAGAB,S.F.On Laguerse polynomials of two variables L n (α,β) (x, y), Bull. Cal. Math. Soc., Vol. 83, pp. 253, (1991). [8] RAINVILLE, E. D. Special Functions, Macmillan, New York; Reprinted by Chelsea Publ. Co., Bronx., New York, (1971). [9] SRIVASTAVA, H. M. AND MANOCHA, H. L. A Treatise on Generating Functions, Ellis Horwood Limited Publishers, Chichester, Halsted Press, a division of John Wiley and Sons, New York, (1984). [10] SRIVASTAVA, H. M. AND KARLSSON, P. W. Multiple Gaussian Hypergeometric Series, John Wiley & Sons (Halsted Press), New York; Ellis Horwood, Chichester, (1985). MUMTAZ AHMAD KHAN Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh - 202002, U. P., India e-mail : mumtaz ahmad khan 2008@yahoo.com AND BAHMAN ALIDAD Department of Mathematics Faculty of Science, Golestan University, Gorgan, Iran e-mail: bahman alidad@yahoo.com