Generalized Watson's summation formula for \(_3F_2(1)\)
type: Item
instance of: scholarly article
title: Generalized Watson's summation formula for \(_3F_2(1)\)
zbMATH Open document ID: 0903.33004
DOI: 10.1016/S0377-0427(97)00170-2
author: Stanisław Lewanowicz
published in: Journal of Computational and Applied Mathematics
publication date: 1997-01-01T00:00:00Z
review text: A summation formula for \({_3F_2}(^{a,b,c}_{d,2c}| 1)\) was given by G. N. Watson and functions of the type \(f_{i,j}(a,b,c)\equiv {_3F_2}(^{a,b,c}_{d+{1\over 2}i,2c+j}| 1)\) were later studied by others. In this paper, the analytical formula \[ f_{i,j}(a,b,c)\equiv P_i(X^{(0)}_\mu T^{(0)}_{ij}- X^{(\ell)}_\mu T^{(\ell)}_{ij}),\;R(2c- a-b)> - i-2j-1, \] where \(\mu=| i|\text{ mod }2\), \(T^{(\ell)}_{i,j}= A_{i,j}Q^{(\ell)}_i+ B_{i,j} R^{(1-\ell)}_i\) \((\ell= 0,1)\), \[ P_i= P_i(a, b,c)= 2^{a+ b+| i|-2}(-1)^{[{| i|\over 2}]}{\Gamma(d+{1\over 2}| i|)\Gamma(c+ {1\over 2}) \Gamma(c- d-{1\over 2}| i|+ 1)\over \Gamma({1\over 2}) \Gamma(a)\Gamma(b)}, \] \[ X^{(\ell)}_\mu= X^{(\ell)}_\mu(a, b,c):= {\Gamma({1\over 2}a+{1\over 2}\ell) \Gamma({1\over 2} b+{1\over 2}\mu+{1\over 2}\ell)\over \Gamma(c-{1\over 2} a+{1\over 2}\ell+{1\over 2}) \Gamma(c-{1\over 2}b- {1\over 2}\mu+{1\over 2}\ell+{1\over 2})}\quad (\ell= 0,1), \] \[ Q^{(\ell)}_i= \sum^{[(| i|- \ell)/2]}_{m= 0} {({1\over 2} a- c+ m)\ell\over ({1\over 2} b+{1\over 2}| i|-{1\over 2}- m)\ell} \alpha_{i,2m+\ell} \beta^{(i)}_{\ell, m}\quad (\ell= 0,1), \] \(\alpha_{ik}\) and \(\beta_{im}^\ell\) have certain values and \(A_{ij}\), \(B_{ij}\) are particular solutions of the difference equation in \(j\) (\(i\) being a parameter) \[ \begin{multlined} (j+ 2c-a)(j+ 2c- b)(j+ c)E_{j+ 1}=\\ (j+ 2c)[(j+ 2c- d)(2j+ 2c- d)+\textstyle{{1\over 2}} i(j+ 2c- 1)- (a- d)_2]E_j-\\ (j+ 2c- 1)_2(j+ c- d+\textstyle{{1\over 2}} i) E_{j- 1} (j\in\mathbb{Z}),\end{multlined} \] obtained for the initial values \(E_0= 1\), \(E_1= 1\), and \(E_0= 0\), \(E_1= 1\) respectively, has been obtained for \(f_{i,j}\) with fixed \(j\) and arbitrary \(i\in\mathbb{Z}\), with the help of a recurrence relation. A corollary is derived and five results derived from the corollary with the help of a program written in Maple, are given as examples. A special case of \({_3F_2}(^{-2n- p,\alpha+ 2n+p,c}_{{1\over 2}\alpha+{1\over 2}i+{1\over 2}, 2c+ j}| 1)\) for \(n= 0,1,\…
Mathematics Subject Classification ID: 33C20
zbMATH DE Number: 1115944
zbMATH Keywords: Watson's summation formula
reviewed by: K. M. Saksena
describes a project that uses: Maple
MaRDI profile type: Publication
cites work: Contiguous Hypergeometric Functions of the Type 3F2(1)
cites work: Q4002478
cites work: Q5820721
cites work: Some Summation Formulas for the Series 3 F 2 (1)
cites work: Notes on a paper by J. B. Miller
cites work: Q4284184
cites work: Generalizations of Whipple's theorem on the sum of a \({}_ 3 F_ 2\)
cites work: Generalizations of Dixon's Theorem on the Sum of A 3 F 2
cites work: Some remarks on uniform asymptotic expansions for Bessel functions
cites work: Q4875364
cites work: Generating Functions for Bessel and Related Polynomials
cites work: Q4453650
cites work: Q5508372
cites work: Irreducible recurrences and representation theorems for \(_ 3F_ 2(1)\)
cites work: Gauss's \(_ 2F_ 1(1)\) cannot be generalized to \(_ 2F_ 1(x)\)
description: scientific article; zbMATH DE number 1115944