Introduction


Abstract


En
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}, which covers certain well-known special functions for particular choices of the parameteres \alpha, \beta and \gamma. For example, for \gamma= 1 and \gamma = -1, \beta = 0 one obtains respectively \psi(\alpha,\beta,1,\chi) = \Gamma(\alpha - \beta; \chi) and \psi(\alpha,0,-1;\chi) = -D(\alpha - 1,\chi),where(Error rendering LaTeX formula) is the complementary incomplete Gamma function and D(𝛼 - 1,x) = \Bigl\lmoustache _{x}<sup>∈fty</sup> dt {{t<sup>𝛼 - 1</sup>} \over {e<sup>t</sup> - 1}} is a function of Debye type.The function \psi(𝛼,𝛽,γ;x) has been introduce by us in order to find a class of exact solutions for nonlinear wave equations of the Klein-Gordon type u_{tx} = ae<sup>-u</sup> + bu<sup>𝛽 - 1</sup>, where u = u(x,t) and a, b are constants.

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