On an extension of Kummer's second theorem (Q369699)

Summary: The aim of this paper is to establish an extension of Kummer's second theorem in the form \(e^{-x/2}\,\,_0F_2 \left[ \begin{matrix} a, & 2+d;\\ 2a+2, & d;\end{matrix} x \right] =\, _0F_1 \left[ \begin{matrix} -;\\ a + 3/2; \end{matrix} x^2/16 \right] + ((a/d - 1/2)/(a + 1))x\,\,_0F_1 \left[ \begin{matrix} -;\\ a + 3/2;\end{matrix} x^2/16\right] + (cx^2/2(2a + 3))\,\,_0F_1 \left[ \begin{matrix} -;\\ a + 5/2; \end{matrix} x^2/16 \right]\), where \(c = (1/(a + 1))(1/2 - a/d) + a/d(d + 1)\), \(d \neq 0, -1, -2, \dots\) For \(d = 2a\), 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 \(_3F_2(2)\) series. The results derived in this paper are simple, interesting, and easily established and may be useful in physics, engineering, and applied mathematics.