Abstract and Applied Analysis

On an Extension of Kummer's Second Theorem

Abstract

The aim of this paper is to establish an extension of Kummer's second theorem in the form $\mathrm{ }{e}^{-x/\mathrm{2}}{F}_{\mathrm{2}}\mathrm{2}[\begin{smallmatrix}a,& \mathrm{2}+d;\\[5pt] x\\[5pt] \mathrm{2}a+\mathrm{2},& d;\end{smallmatrix}]=F\mathrm{1}\mathrm{0}[\begin{smallmatrix}-;\\[5pt] {x}^{\mathrm{2}}/\mathrm{16}\\[5pt] a+\mathrm{3}/\mathrm{2};\end{smallmatrix}]+((a/d-\mathrm{1}/\mathrm{2})/(a+\mathrm{1}))x {F}_{\mathrm{1}}\mathrm{0}[\begin{smallmatrix}-;\\[5pt] {x}^{\mathrm{2}}/\mathrm{16}\\[5pt] a+\mathrm{3}/\mathrm{2};\end{smallmatrix}]+(c{x}^{\mathrm{2}}/\mathrm{2}(\mathrm{2}a+\mathrm{3})){F}_{\mathrm{1}}\mathrm{0}[\begin{smallmatrix}-;\\[5pt] {x}^{\mathrm{2}}/\mathrm{16}\\[5pt] a+\mathrm{5}/\mathrm{2};\end{smallmatrix}]$, where $c=(1/(a+1))(1/2-a/d)+a/d(d+1), d\ne 0,-1,-2,\dots$. For $d=\mathrm{2}a$, we recover Kummer's second theorem. The result is derived with the help of Kummer's second theorem and its contiguous results available in the literature. As an application, we obtain two general results for the terminating ${F}_{2}3(2)$ series. The results derived in this paper are simple, interesting, and easily established and may be useful in physics, engineering, and applied mathematics.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 128458, 6 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393448855

Digital Object Identifier
doi:10.1155/2013/128458

Mathematical Reviews number (MathSciNet)
MR3039181

Zentralblatt MATH identifier
1277.33008

Citation

Rakha, Medhat A.; Awad, Mohamed M.; Rathie, Arjun K. On an Extension of Kummer's Second Theorem. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 128458, 6 pages. doi:10.1155/2013/128458. https://projecteuclid.org/euclid.aaa/1393448855