Degenerate Analogues of Euler Zeta, Digamma, and Polygamma Functions

In recent years, much attention has been paid to the role of degenerate versions of special functions and polynomials in mathematical physics and engineering. In the present paper, we introduce a degenerate Euler zeta function, a degenerate digamma function, and a degenerate polygamma function. We present several properties, recurrence relations, inﬁnite series, and integral representations for these functions. Furthermore, we establish identities involving hypergeometric functions in terms of degenerate digamma function.


Introduction
e gamma, digamma, and polygamma functions have an increasing and recognized role in fractional differential equations, mathematical physics, the theory of special functions, statistics, probability theory, and the theory of infinite series. e reader may refer, for example, to [1][2][3][4][5][6][7][8][9]. ese functions are directly connected with a variety of special functions such as zeta function, Clausen's function, and hypergeometric functions. e evaluations of series involving Riemann zeta function ζ(s) and related functions have a long history that can be traced back to Christian Goldbach (1690-1764) and Leonhard Euler (1707-1783) (see, for details, [10]). e Euler zeta function and its generalizations and extensions have been widely studied [11][12][13][14][15].
Motivated by this great importance of these functions, their investigations and generalizations to the degenerate framework have been widely considered in the literature, for instance, [24][25][26][27].
In this section, we present some basic properties and well-known results on a degenerate gamma function which we need in this work. In Section 2, we introduce a degenerate Euler zeta function and discuss its region of convergence, integral representation, and infinite series representation. In Section 3, we define a degenerate digamma function along with its region of convergence and integral representation. We also give certain recurrence relations and formulae satisfied by the degenerate digamma function. In Section 4, we define a degenerate polygamma function and describe its convergence conditions. Some recurrence relations satisfied by the degenerate polygamma function are also given here. Finally, in Section 5, the hypergeometric functions are expressed in terms of the degenerate digamma function.
In [26], a degenerate gamma function, denoted Γ * λ , has been defined by e basic results of this function, given in [26], can be summarized in the following lemma.
Also, we can easily show that where Γ(z) is the gamma function. Moreover, for m, n ∈ N, we have where B(., .) is the beta function.

Degenerate Euler Zeta Function
e Euler zeta function in two complex variables s, z such that Re(s) > 0 and Re(z) > 0 is defined by (see [12,24]) An integral representation of ζ E (s, z) is given as where where E n (z) is the Euler polynomial of degree n. When z � 0, E n � E n (0) are Euler numbers (see, [12,14]). Kim in [14] obtained that ζ n (− n, z) � E n (z), n ≥ 0.
In this section, we consider a degenerate analogue of the Euler zeta function which is given as where λ ∈ (0, 1), s, z ∈ C with Re(s) > 0, Re(z) > 0, and By (9) and (11), it follows that which is the degenerate Euler polynomial of degree n.
Furthermore, from (11), it follows Hence, we obtain the following results.

Degenerate Digamma Function
e digamma function, denoted by ψ(z), is the logarithmic derivative of the gamma function given by [6,16,28]: In this section, we define a degenerate digamma function as follows: where Γ * λ (z) is the degenerate gamma function defined by (1). Now, we are going to obtain certain functional equations involving the degenerate digamma function ψ * λ (z). Using (2) and (21), it follows that Generally, we have the following.

Theorem 4.
For z ∈ C, Re(z) > 0, and λ ∈ (0, 1), Next, the degenerate digamma function ψ * λ (z) defined by (21) can be expressed as a series expression in terms of Riemann's zeta function. Using equation (29) can be rewritten as us, one gets the following.
Using the Legendre duplication formula [29] and (5), one can simply find Equation (35) can be extended to an arbitrary integral multiplication of z as follows.
Remark 3. Its worth to mention here that all plotted functions in the below figures were multiplied by sin x, since Fourier space, for the sake of clarify the results to the reader. Now, we are going to find the integral representations for the degenerate digamma function ψ * λ (z), defined by (21), as follows. Note that Hence, using (28) and (37), it can be shown that Now, substituting t � (1 + λ) − s/λ in (37) gives Since and by integrating from 1 to n, it follows that Inserting (41) and in (27), we get Since the last limit equals to zero, it follows e following theorem summarizes the above results.

Degenerate Polygamma Function
e polygamma function of order m is obtained by taking the (m + 1)th derivative of the logarithm of gamma function (cf. [28]). us, In this section, we define the degenerate polygamma function of order m as 4 Mathematical Problems in Engineering

Mathematical Problems in Engineering
where Γ * λ (z) is the degenerate gamma function defined by (1) and ψ * λ (z) is the degenerate digamma function defined by (21).