engleski

On a summation formula for the Clausen's series ${;;};;_3F_2$ with applications

The aim of this research paper is to establish the following summation formula for the Clausen's series ${; ; }; ; _3F_2$ viz. \[ {; ; }; ; _3F_2\Bigg[ \begin{; ; array}; ; {; ; c}; ; -n, \, \, \, b-a-1, \, \, \, f+1\\ \\b, \, \, \, f \end{; ; array}; ; 1\Bigg] = \frac{; ; (a)_n(c+1)_n}; ; {; ; (b)_n(c)_n}; ; \] where $f= c(1+a-b)/(a-c)$, in {; ; \em three different ways}; ; ; . For $c=\tfrac12 a$, we have \[ {; ; }; ; _3F_2\Bigg[ \begin{; ; array}; ; {; ; c}; ; -n, \, \, \, b-a-1, \, \, \, 2+a-b\\ \\b, \, \, \, 1+a-b \end{; ; array}; ; 1\Bigg] = \frac{; ; (a)_n\big(1+\tfrac12 a\big)_n}; ; {; ; (b)_n\big( \tfrac12 a\big)_n}; ; \] which is already available in the literarute. Our formula is then applied to obtain two general results, one is the Euler's transformation for the series ${; ; }; ; ; 2F_2$ and another is the Kummer-type first transformation for the series ${; ; }; ; _2F_2$ established recently by Paris by following a different method. The results obtained generalize the related results by Exton.

Clausen's series ${; ; }; ; _3F_2$ ; Euler's transformation for ${; ; }; ; _2F_2$ ; Kamp\'e de F\'eriet function ; Kummer-type I transformation for ${; ; ; }; ; ; _2F_2$

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