Abstract
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.
1. Introduction
The 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. The reader may refer, for example, to [1–9]. These functions are directly connected with a variety of special functions such as zeta function, Clausen’s function, and hypergeometric functions. The evaluations of series involving Riemann zeta function 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]). The Euler zeta function and its generalizations and extensions have been widely studied [11–15].
Later on, these functions arise in the study of matrix-valued special functions and in the theory of matrix-valued orthogonal polynomials, see e.g., [16–23] and the references therein.
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, [24–27].
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 , has been defined by
The basic results of this function, given in [26], can be summarized in the following lemma.
Lemma 1. Let . Then, for with satisfiesAlso, we can easily show that
Corollary 1. Let . Then, satisfieswhere is the gamma function. Moreover, for , we havewhere is the beta function.
2. Degenerate Euler Zeta Function
The Euler zeta function in two complex variables such that and is defined by (see [12, 24])
An integral representation of is given aswherewhere is the Euler polynomial of degree . When , are Euler numbers (see, [12, 14]). Kim in [14] obtained that .
In this section, we consider a degenerate analogue of the Euler zeta function which is given aswhere with , and
By (9) and (11), it follows thatwhich is the degenerate Euler polynomial of degree .
From (10) and (11), we obtain that
Thus, using (10) and (13), we conclude the following result.
Theorem 1. For with , and , the degenerate Euler zeta function defined in (10) has the following infinite series representation:Moreover, in view of (5), we havewhere is the Euler zeta function defined by (7).
Furthermore, from (11), it follows
Hence, we obtain the following results.
Theorem 2. For with , and , the degenerate Euler zeta function , defined in (10), satisfiesAnd for ,
Remark 1. Note that is an entire function in the complex plane.
Remark 2.
3. Degenerate Digamma Function
The digamma function, denoted by , 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 is the degenerate gamma function defined by (1). Now, we are going to obtain certain functional equations involving the degenerate digamma function . Using (2) and (21), it follows that
Generally, we have the following.
Theorem 3. For , , we have
Furthermore, using relation (5), we find thatwhere is the digamma function defined by (20). According to Batir [28], we havewhereis the Euler–Mascheroni constant. Hence, substituting (25) into (24), one gets the following.
Theorem 4. For , and ,
Next, the degenerate digamma function defined by (21) can be expressed as a series expression in terms of Riemann’s zeta function. Usingequation (29) can be rewritten as
Thus, one gets the following.
Theorem 5. For , and ,
Note that these series converge absolutely for .
Using the Legendre duplication formula [29]and (5), one can simply find
Equation (35) can be extended to an arbitrary integral multiplication of as follows.
Theorem 6. For , and ,
Figures 1–3 illustrate the degenerate digamma function in (24) at different values for .
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Remark 3. Its worth to mention here that all plotted functions in the below figures were multiplied by , 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 , defined by (21), as follows. Note thatHence, using (28) and (37), it can be shown thatNow, substituting in (37) givesSinceand by integrating from 1 to , it follows thatInserting (41) andin (27), we getSince the last limit equals to zero, it followsThe following theorem summarizes the above results.
Theorem 7. For , the degenerate digamma function , defined by (10), can be expressed as (38), (39) as well as (44).
4. Degenerate Polygamma Function
The polygamma function of order is obtained by taking the th derivative of the logarithm of gamma function (cf. [28]). Thus,
In this section, we define the degenerate polygamma function of order aswhere is the degenerate gamma function defined by (1) and is the degenerate digamma function defined by (21).
By (24), it follows that
Using (44), an integral representation for , given in the next theorem, can be obtained.
Theorem 8. Let and . Then, for with , the degenerate polygamma function , defined by (46), can be expressed asThe following recurrence relations for the degenerate polygamma function defined by (47) can be obtained from (22)–(24), (35), and (36) as the following.
Theorem 9. For , and , the recurrence relations hold true:From (25), a series representation of the degenerate polygamma function is given in the following result.
Theorem 10. For , and , we have
Remark 4. The degenerate polygamma function can be expressed in terms of the generalized zeta functionasFinally, using (32), a series representation in terms of the Riemann zeta function can be obtained, see the following result.
Theorem 11. For , and , we have
5. Applications
Let with and . Then, it can be verified that
Now, we can directly use the integral transform of Gauss hypergeometric function (see [29]) and the formulae:
Using (54) in (56) and L’Hôpital rule for complex numbers with applying equation (24) yields the following identity in terms of the degenerate digamma function:
Similarly, we can present another identity involving hypergeometric function in terms of our degenerate digamma function in the following form:
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All the authors contributed equally and significantly to writing this article. All the authors read and approved the final manuscript.
Acknowledgments
The authors wish to acknowledge the approval and the support of this research study from the Deanship of Scientific Research in King Khalid University, Abha, Saudi Arabia, under grant R.G.P-79-41. The first and second authors wish to acknowledge the approval and the support of this research study from the National Natural Science Foundation of China (11601525). The authors are thankful to Professor Hillal Elshehabey for the figures illustrated in this paper.