Geodesic
Generation of Kummer's second theorem with application
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 3, pp. 407-422 Cet article a éte moissonné depuis la source Math-Net.Ru

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The aim of this research paper is to obtain single series expression of $$ e^{-x/2}{}_1F_1(\alpha; 2\alpha+i; x) $$ for $i=0$, $\pm1$, $\pm2$, $\pm3$, $\pm4$, $\pm5$, where ${}_1F_1(\cdot)$ is the function of Kummer. For $i=0$, we have the well known Kummer second theorem. The results are derived with the help of generalized Gauss second summation theorem obtained earlier by Lavoie et al. In addition to this, explicit expressions of $$ {}_2F_1[-2n, \alpha; 2\alpha+i; 2] \text{ and } {}_2F_1[-2n-1, \alpha; 2\alpha+i; 2] $$ each for $i=0$, $\pm1$, $\pm2$, $\pm3$, $\pm4$, $\pm5$ are also given. For $i=0$, we get two interesting and known results recorded in the literature. As an applications of our results, explicit expressios of $$ e^{-x}{}_1F_1(\alpha; 2\alpha+i; x)\times{}_1F_1(\alpha; 2\alpha+j; x) $$ for $i$, $j=0$, $\pm1$, $\pm2$, $\pm3$, $\pm4$, $\pm5$ and $$ (1-x)^{-a}{}_2F_1\left(a, b; 2b+j; -\frac{2x}{1-x}\right) $$ for $j=0$, $\pm1$, $\pm2$, $\pm3$, $\pm4$, $\pm5$ are given. For $i=j=0$ and $j=0$, we respectively get the well known Preece identity and a well known quadratic transformation formula due to Kummer. The results derived in this paper are simple, interesting, easily established and may by useful in the applicable sciences. 
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Yong Sup Kim; M. A. Rakha; A. K. Rathie. Generation of Kummer's second theorem with application. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 3, pp. 407-422. http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_3_a1/

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