The Representations on the Partial Derivatives of the Extended, Generalized Gamma and Incomplete Gamma Functions and Their Applications

In this paper, some recursive relations of the derivatives of the gamma function Γ(α) and incomplete gamma function Γ(α, z) are obtained for complex number α ̸= 0,−1,−2, · · ·. Moreover, the recurrence relation of the derivative of the k−gamma function Γk(x) are also given. Thus, the partial derivative ∂ p+q ∂νp∂μq B(ν, μ) of the Beta function B(ν, μ) can also be represented for p, q = 0, 1, 2, · · ·. Based on these results, the partial derivatives of the extended, generalized complete and incomplete gamma functions are obtained. Furthermore, the partial derivative of confluent hypergeometric function is also considered.