Some new class of special functions suggested by the confluent hypergeometric function

Abstract

In the present work, we introduce the function representing a rapidly convergent power series which extends the well-known confluent hypergeometric function $_1F_1[z]$ as well as the integral function $f(z) = \sum \nolimits _{n=1}^\infty \frac{z^n}{n!^n}$ considered by Sikkema (Differential operators and equations, P. Noordhoff N. V., Djakarta, 1953). We introduce the corresponding differential operators and obtain infinite order differential equations, for which these new special functions are the eigen functions. First we establish some properties, as the order zero of these entire (integral) functions, integral representations, differential equations involving a kind of hyper-Bessel type operators of infinite order. Then we emphasize on the special cases, especially the corresponding analogues of the exponential, circular and hyperbolic functions, called here as ${\ell }$-H exponential function, ${\ell }$-H circular and ${\ell }$-H hyperbolic functions. At the end, the graphs of these functions are plotted using the Maple software.