Generalized Watson's summation formula for \(_3F_2(1)\)
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 \(n= 0,1,\dots\); \(p= 0,1\) and \(i\), \(j\in\mathbb{Z}\) is mentioned and it is shown that for any \(i\), \(j\in \mathbb{Z}\), \[ g_{i,j}(a, c,e)= {\Gamma(e)\Gamma(1+ i+ 2c- e)\Gamma(c)\over \Gamma(a)\Gamma(1+ i- a+ c)\Gamma(2c- j)} f_{i,j}(e- a, 1+ i+ 2c- a- e,c-j) \] and a formula implied by the above results is given. A theorem, followed by an example, is also proved to show that the computation of \({_3F_2}(^{a,b,c}_{1+ i+ a-b, 1+ i+ j+ a-c}| 1)\) for integral \(i\), \(j\) can be reduced to the evaluation of \(f_{i,j}\). A collection of forms for \(A_{i,j}\) and \(B_{i,j}\) (\(j\in\mathbb{Z}\) and \(-2\leq j\leq 3\)) is given at the end.
scientific article; zbMATH DE number 1115944