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

Some new class of special functions suggested by the confluent hypergeometric function In the present work, we introduce the function representing a rapidly convergent power series which extends the well-known confluent hypergeometric function $$_1F_1[z]$$ 1 F 1 [ z ] as well as the integral function $$ f(z) = \sum \nolimits _{n=1}^\infty \frac{z^n}{n!^n} $$ f ( z ) = ∑ n = 1 ∞ 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 $${\ell }$$ ℓ -H exponential function, $${\ell }$$ ℓ -H circular and $${\ell }$$ ℓ -H hyperbolic functions. At the end, the graphs of these functions are plotted using the Maple software. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ANNALI DELL'UNIVERSITA' DI FERRARA Springer Journals

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

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Publisher
Springer Journals
Copyright
Copyright © 2015 by Università degli Studi di Ferrara
Subject
Mathematics; Mathematics, general; Analysis; Geometry; History of Mathematical Sciences; Numerical Analysis; Algebraic Geometry
ISSN
0430-3202
eISSN
1827-1510
DOI
10.1007/s11565-015-0238-3
Publisher site
See Article on Publisher Site

Abstract

In the present work, we introduce the function representing a rapidly convergent power series which extends the well-known confluent hypergeometric function $$_1F_1[z]$$ 1 F 1 [ z ] as well as the integral function $$ f(z) = \sum \nolimits _{n=1}^\infty \frac{z^n}{n!^n} $$ f ( z ) = ∑ n = 1 ∞ 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 $${\ell }$$ ℓ -H exponential function, $${\ell }$$ ℓ -H circular and $${\ell }$$ ℓ -H hyperbolic functions. At the end, the graphs of these functions are plotted using the Maple software.

Journal

ANNALI DELL'UNIVERSITA' DI FERRARA – Springer Journals

Published: Dec 30, 2015

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