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Further generalization of Kobayashi’s gamma function. (English) Zbl 0994.33001

In 1991, K. Kobayashi [J. Phys. Soc. Japan 60, 1501-1512, 1881-1905 (1991)] introduced the following generalization of the gamma function \(\Gamma(u)\), namely: \[ \Gamma_m (u,v) = \int_0^{\infty} {\frac {t^{u-1} e^{-t}} {(t+v)^{m}}} dt, \quad \text{where} m \;\;\text{is a positive integer}, \tag{1} \] which is essentially a confluent hypergeometric function of the second kind. Kobayashi’s function has found important applications in the wave scattering and diffraction theory in relation to the Wiener-Hopf technique but restricted to the case where obstacles have semi-infinite boundary. For possible treatment of some diffraction problems with obstacles of different geometries, F. Al-Musallam and S. L. Kalla [Appl. Anal. 66, No. 1-2, 173-187 (1997; Zbl 0886.33002)] generalized further (1) to the form \[ D \binom{a,b,c,p}{u,v} = v^{-a} \int_0^{\infty} t^{u-1} e^{-pt} {}_2F_1 \left(a,b;c; -\tfrac tv\right) dt, \tag{2} \] where the Gauss hypergeometric function \({}_2F_1\) is involved in the integrand.
In this article, the authors introduce further generalization of the gamma function and of the functions (1), (2): \[ D \binom{a,b,c,p}{u,v,\delta}= v^{-a} \int_0^{\infty} t^{u-1} \left(1 - {\tfrac tv}\right)^{\delta-1} e^{-pt} {}_2F_1 \left(a,b;c; -\tfrac tv\right) dt, \tag{3} \] where Re\((u)>0\), Re\((p)>0, |\arg v|<\pi, \;b, a, c\) are complex parameters with \(c \neq 0,-1,-2,\dots\). Note that for \(\delta=1\), (3) reduces to (2), and additionally, if \(p=1, b=c, a=m\) (\(m\) a positive integer), it becomes the Kobayashi function (1).
The authors introduce also the respective counterparts of the incomplete and complementary gamma functions defined by integrals \(\int_0^w\) and \(\int_w^{\infty}\) of the form (3). They study some properties of these functions: differential relations, recurrence relations, and asymptotic series expansions of (3) as \(v \to \infty\).

MSC:

33B15 Gamma, beta and polygamma functions
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)

Citations:

Zbl 0886.33002
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