On Properties of an Entire Function That is a Generalization of the Wright Function
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ON PROPERTIES OF AN ENTIRE FUNCTION THAT IS A GENERALIZATION OF THE WRIGHT FUNCTION L. L. Karasheva
UDC 517.58
Abstract. In this paper, we examine properties of an entire function that is a generalization of the Wright function. Various representations, estimates, and differentiation formulas are obtained. Keywords and phrases: integer function, Wright function, special function. AMS Subject Classification: 33E20, 33E99
1.
Introduction. Let us consider the function ∞ C,n Θn,α (z; μ) = =0
where C,n =
z ! Γ μ −
α 2n
,
(1)
sin (1+)π 2
. sin (1+)π 2n It is easy to see that in the case where + 1 is multiple of 2n, the numerator and the denominator in the expression of C,n vanish and C2nk−1,n = lim
ζ→πk
sin ζn = n(−1)k(n+1) , sin ζ
k ∈ Z, n ∈ N.
Moreover, n−1
sin ζn iζ(2k−n+1) = e , sin ζ
(2)
k=0
so that, obviously, |C,n | ≤ n for all ∈ N ∪ {0}. Consequently, the series (1) defines an entire function of the variable z ∈ C depending on the parameters n ∈ N, α < 2n, and μ ∈ C. A function of the form (1) for n = 1 coincides with the Wright function (see [9]) α Θ1,α (z; μ) = φ − , μ; z , 2 which is involved in the expression of the fundamental solution of the diffusion-wave equation (see [8]). The main interest in the study of the function (1) is due to the fact that this function is involved in the fundamental solution of the following equation (see [3]): 2n ∂α n∂ u u(x, t) + (−1) = f (x, t), (3) ∂tα ∂x2n where n ∈ N and ∂ α /∂tα is the fractional derivative of order α. We note that in terms of the function (1), fundamental solutions of higher-order parabolic equations can be also expressed in the case of integer α < 2n.
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 149, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, 2018. c 2020 Springer Science+Business Media, LLC 1072–3374/20/2505–0753
753
This paper is devoted to the study of properties of the function (1); in particular, we obtain for it representations in the form of a sum of Wright functions, an improper integral, the Fox H-function, a contour integral, and formulas and estimates for integer and fractional differentiation. 2. Representation in terms of the Wright function. For any z ∈ C and α < 1, the following representation holds for the function Θn,α (z; μ): Θn,α (z; μ) =
n−1
ei
(2k−n+1)π 2n
k=0
α (2k−n+1)π φ − , μ; zei 2n , 2n
(4)
where φ(α, β; z) is the Wright function (see [9]). Using Eq. (2), we rewrite the function Θn,α (z; μ) in the following form: Θn,α (z; μ) =
∞ n−1
ei
(2k−n+1)(1+)π 2n
=0 k=0
=
∞ n−1
z ! Γ μ −
α 2n
i (2k−n+1)π+(2k−n+1)π 2n
e
=0 k=0
z !Γ μ −
= α
2n
∞ n−1
i (2k−n+1)π 2n
e
=0 k=0
(2k−n+1)π
(zei 2n !Γ μ −
) . α
2n
Taking into account the representation of the Wright function in the series form, we rewrite the last equation in the form Θn,α (z; μ) =
n−1
ei
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