Introduction
Abstract
En
In this paper, some basic properties (recurrence relations,asymptotic expansion, series representation and others) are derived for the new function
, which covers certain well-known special functions for particular choices of the parameteres
. For example, for
and
,
one obtains respectively
and
,where(Error rendering LaTeX formula) is the complementary incomplete Gamma function and
is a function of Debye type.The function
has been introduce by us in order to find a class of exact solutions for nonlinear wave equations of the Klein-Gordon type
, where
and a, b are constants.
In this paper, some basic properties (recurrence relations,asymptotic expansion, series representation and others) are derived for the new function
![\psi(\alpha,\beta,\gamma,\chi) = \lmoustache _{x}^ \infty {dt t^\alppha-1 [1-(1- {{e^-t} \frac {t^\beta}})^y}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/1ddd2feb3286bb68d56aa29746c367f7.png)
![\alpha, \beta and \gamma](http://siba-ese.unile.it/plugins/generic/latexRender/cache/d4f42f35a37841451659539fa7577fb0.png)
![\gamma= 1](http://siba-ese.unile.it/plugins/generic/latexRender/cache/a9980b2fbd8e2e2acbc3fe853749f122.png)
![\gamma = -1](http://siba-ese.unile.it/plugins/generic/latexRender/cache/a143930ff1bda05e259e68283814a5b0.png)
![\beta = 0](http://siba-ese.unile.it/plugins/generic/latexRender/cache/6ba4d761cd412e149092cb31b4ca6374.png)
![\psi(\alpha,\beta,1,\chi) = \Gamma(\alpha - \beta; \chi)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/f60e03660e6b8f17d0b5ffdc1fae142e.png)
![\psi(\alpha,0,-1;\chi) = -D(\alpha - 1,\chi)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/008044c8cee90e330f6a90a2d2d85fd3.png)
![D(𝛼 - 1,x) = \Bigl\lmoustache _{x}<sup>∈fty</sup> dt {{t<sup>𝛼 - 1</sup>} \over {e<sup>t</sup> - 1}}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/90c1382267500dcd535d089d4580f63e.png)
![\psi(𝛼,𝛽,γ;x)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/9c100de0168a30ff3fc55e6576b0023e.png)
![u_{tx} = ae<sup>-u</sup> + bu<sup>𝛽 - 1</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/24a1168ea2687d2d4dac58849a5fc69d.png)
![u = u(x,t)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/f6c9681ec2b2e6f2deb5654553fb7438.png)
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