Expansion of the confluent hypergeometric function in series of Bessel functions

Abstract:

An expansion of the confluent hypergeometric function $\Phi (a,c,z)$ in series of functions of the same kind has been given by Buchholz [1]. By specialization of some quantities, there is obtained an expansion in series of modified Bessel functions of the first kind, ${I_\upsilon }(z)$, where $v$ depends on the parameter $a$. Tricomi [2, 3] has developed two expansions of similar type where both the order and argument of the Bessel functions depend on the parameters $a$ and $c$. In the present paper, we derive an expansion in series of Bessel functions of integral order whose argument is independent of $a$ and $c$. Our expansion is advantageous for many purposes of computation since the parameters and variable of $\Phi (a,c,z)$ appear in separated form. Also, for desk calculation, extensive tables of ${I_n}(z)$ are available, while for automatic computation Bessel functions are easy to generate [4]. Special cases of the confluent function, such as the incomplete gamma function, are also studied. For the class of transcendents known as the error functions, including the Fresnel integrals, it is shown that our expansion coincides with that of Buchholz [1]. By specializing a parameter and passing to a limit, we derive expansions for the exponential integral and related functions. Other expansions for the error and exponential integrals are derived on altogether different bases. Finally, some numerical examples are presented to manifest the efficiency of our formulas.