# Degenerate Analogues of Euler Zeta, Digamma, and Polygamma Functions

Authors: Fuli He; Ahmed Bakhet; Mohamed Akel; Mohamed Abdalla
Journal: Mathematical Problems in Engineering
 (2020)

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## 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.


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## Body

## 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 ζs 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 Section2, 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 Γλ∗, has been defined by(1)Γλ∗z=∫0∞1+λ−t/λtz−1dt,λ∈0,1,Rez>0.The basic results of this function, given in [26], can be summarized in the following lemma.Lemma 1.
Letλ∈0,1. Then, for z∈ℂ with Rez>0,Γλ∗z satisfies(2)Γλ∗z+1=λzlog1+λΓλ∗z,Γλ∗1=λlog1+λ,(3)Γλ∗z+1=λk+1zz−1⋯z−klog1+λk+1Γλ∗z−k,k≥0,(4)Γλ∗k+1=λk+1k!log1+λk+1,k∈ℕ.
Also, we can easily show thatCorollary 1.
Letλ∈0,1. Then, Γλ∗z satisfies(5)Γλ∗z=λlog1+λzΓz,z∈ℂ,Rez>0,where Γz is the gamma function. Moreover, for m,n∈N, we have(6)Γλ∗mΓλ∗n=Bm,nΓλ∗m+n,where B.,. is the beta function.


## 2. Degenerate Euler Zeta Function

The Euler zeta function in two complex variabless,z such that Res>0 and Rez>0 is defined by (see [12, 24])(7)ζEs,z=2∑n=0∞−1nn+zs.An integral representation ofζEs,z is given as(8)ζEs,z=Γ−1s∫0∞F−t,zts−1dt,where(9)Ft,z=2ezt1+et=∑n=0∞Enztnn!,where Enz is the Euler polynomial of degree n. When z=0, En=En0 are Euler numbers (see, [12, 14]). Kim in [14] obtained that ζn−n,z=Enz,n≥0.In this section, we consider a degenerate analogue of the Euler zeta function which is given as(10)ζEλs,z=Γ−1s∫0∞Fλ−t,zts−1dt,where λ∈0,1,s,z∈ℂ with Res>0,Rez>0, and(11)Fλt,z=21+1+λt/λ1+λzt/λ=∑n=0∞ℰnλztnn!.By (9) and (11), it follows that(12)ℰnλz=λln1+λnEnz,which is the degenerate Euler polynomial of degree n.From (10) and (11), we obtain that(13)Γ−1s∫0∞Fλ−t,zts−1dt=Γ−1s∫0∞2∑m=0∞−1m1+λ−m+zt/λts−1dt=2Γ−1s∑m=0∞−1m∫0∞1+λ−τ/λτs−1m+zsdτ=2Γλ∗sΓs∑m=0∞−1m1m+zs.Thus, using (10) and (13), we conclude the following result.Theorem 1.
Fors,z∈ℂ with Res>0,Rez>0, and λ∈0,1, the degenerate Euler zeta function ζEλs,z defined in (10) has the following infinite series representation:(14)ζEλs,z=2Γλ∗sΓs∑m=0∞−1m1m+zs.
Moreover, in view of (5), we have(15)ζEλs,z=ζEs,zλln1+λs,where ζEs,z is the Euler zeta function defined by (7).Furthermore, from (11), it follows(16)Γ−1s∫0∞Fλ−t,zts−1dt=Γ−1s∑m=0∞ℰnλz−1mm!∫0∞ts+m−1dt.Hence, we obtain the following results.Theorem 2.
Fors,z∈ℂ with Res>0,Rez>0, and λ∈0,1, the degenerate Euler zeta function ζEλs,z, defined in (10), satisfies(17)ζEλs,z=Γ−1s∑m=0∞ℰnλz−1mm!∫0∞ts+m−1dt.
And forn∈ℕ∪0,(18)ζEλ−n,z=2πi−1nn!Γ−nℰnλ=ℰnλz.Remark 1.
Note thatζEλs,z is an entire function in the complex s−plane.Remark 2.
(19)
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## 3. Degenerate Digamma Function

The digamma function, denoted byψz, is the logarithmic derivative of the gamma function given by [6, 16, 28]:(20)ψz=ddzlogΓz=Γ′zΓz.In this section, we define a degenerate digamma function as follows:(21)ψλ∗z=ddzlogΓλ∗z=Γλ∗′zΓλ∗z,where Γλ∗z is the degenerate gamma function defined by (1). Now, we are going to obtain certain functional equations involving the degenerate digamma function ψλ∗z. Using (2) and (21), it follows that(22)ψλ∗z+1=Γλ∗′z+1Γλ∗z+1=zΓλ∗z′zΓλ∗z=Γλ∗′zΓλ∗z+1z=ψλ∗z+1z,Rez>0.Generally, we have the following.Theorem 3.
Forn∈ℕ, z∈ℂ,andRez>0, we have(23)ψλ∗z+n=ψλ∗z+∑m=0n−11z+m.Furthermore, using relation (5), we find that(24)ψλ∗z=ψz+logλlog1+λ,where ψ is the digamma function defined by (20). According to Batir [28], we have(25)ψz=−γ+∑n=0∞z−1n+1n+z,where(26)γ=limn⟶∞∑k=1n1k−logn=−0.577215is the Euler–Mascheroni constant. Hence, substituting (25) into (24), one gets the following.Theorem 4.
Forz∈ℂ,Rez>0, and λ∈0,1,(27)ψλ∗z=logλlog1+λ−limn⟶∞logn−∑j=0n1z+j,(28)ψλ∗z=logλlog1+λ−γ+z−1∑n=0∞1n+1n+z,(29)ψλ∗z+1=logλlog1+λ−γ+z∑n=1∞1nn+z.Next, the degenerate digamma functionψλ∗z defined by (21) can be expressed as a series expression in terms of Riemann’s zeta function. Using(30)n+z−1=n−1∑m=0∞−znm,equation (29) can be rewritten as(31)ψλ∗z+1=logλlog1+λ−γ−∑n=1∞∑m=1∞n−m+1−zm.Thus, one gets the following.Theorem 5.
Forz∈ℂ,Rez>0, and λ∈0,1,(32)ψλ∗z+1=logλlog1+λ−γ−∑m=1∞ζm+1−zm.Note that these series converge absolutely forz<1.Using the Legendre duplication formula [29](33)Γ12Γ2z=22z−1ΓzΓz+12and (5), one can simply find(34)Γλ∗12Γλ∗2z=22z−1Γλ∗zΓλ∗z+12,(35)ψλ∗2z=log2+12ψλ∗z+12ψλ∗z+12,Rez>0.Equation (35) can be extended to an arbitrary integral multiplication of z as follows.Theorem 6.
Forz∈ℂ,Rez>0, and λ∈0,1,(36)ψλ∗mz=logm+1m∑j=1mψλ∗z+j−1m,Rez>0.Figures1–3 illustrate the degenerate digamma function ψλ∗z in (24) at different values for λ∈0,1.Figure 1
Absolute plots of the degenerate digamma function. (a)λ = 0.1. (b) λ = 0.5. (c) λ = 1.0.
(a)
(b)
(c)Figure 2
Real-part plots of the degenerate digamma function. (a)λ = 0.1. (b) λ = 0.5. (c) λ = 1.0.
(a)
(b)
(c)Figure 3
Imagery-part plots of the degenerate digamma function. (a)λ = 0.1. (b) λ = 0.5. (c) λ = 1.0.
(a)
(b)
(c)Remark 3.
Its worth to mention here that all plotted functions in the below figures were multiplied bysinx, 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 functionψλ∗z, defined by (21), as follows. Note that(37)∫011−tz−11−t−1dt=∑n=0∞∫011−tz−1tndt=z−1∑n=0∞1n+1n+z.
Hence, using (28) and (37), it can be shown that(38)ψλ∗z=−γ+logλlog1+λ+∫011−tz−11−t−1dt.
Now, substitutingt=1+λ−s/λ in (37) gives(39)ψλ∗z=−γ+logλlog1+λ+log1+λλ×∫0∞1+λ−t/λ−1+λ−zt/λ1−1+λ−t/λ−1dt.
Since(40)z−1=log1+λλ∫0∞1+λ−zt/λdtand by integrating from 1 to n, it follows that(41)logn=∫0∞∫1n1+λ−zt/λ⋅log1+λ1/λdzdt=∫0∞∫1n1tdz1+λ−zt/λdt=∫0∞1t1+λ−t/λ−1+λ−nt/λdt.
Inserting (41) and(42)z+j−1=log1+λλ∫0∞1+λ−z+jt/λdtin (27), we get(43)ψλ∗z=logλlog1+λ+limn⟶∞∫0∞1+λ−t/λ−1+λ−nt/λ1t−∑j=0nlog1+λλ1+λ−z+jt/λdt=logλlog1+λ+limn⟶∞∫0∞1+λ−tλt−1−log1+λλ1+λ−ztλ1−1+λ−tλ−1dt−limn⟶∞∫0∞1+λ−ntλt−1−log1+λλ1+λ−ztλ1−1+λ−tλ−1dt.
Since the last limit equals to zero, it follows(44)ψλ∗z=logλlog1+λ+∫0∞1t1+λ−t/λ−log1+λλ1−1+λ−t/λ−11+λ−zt/λdt.
The following theorem summarizes the above results.Theorem 7.
Forz∈ℂ,Rez>0,andλ∈0,1, the degenerate digamma function ψλ∗z, defined by (10), can be expressed as (38), (39) as well as (44).


## 4. Degenerate Polygamma Function

The polygamma function of orderm is obtained by taking the m+1th derivative of the logarithm of gamma function (cf. [28]). Thus,(45)ψmz=dmdzmψz=dm+1dzm+1logΓz,Rez>0.In this section, we define the degenerate polygamma function of orderm as(46)ψλ∗mz=dmdzmψλ∗z=dm+1dzm+1logΓλ∗z,Rez>0,where Γλ∗z is the degenerate gamma function defined by (1) and ψλ∗z is the degenerate digamma function defined by (21).By (24), it follows that(47)ψλ∗mz=ψmz,Rez>0.Using (44), an integral representation for ψλ∗mz, given in the next theorem, can be obtained.Theorem 8.
Letλ∈0,1 and m∈ℕ. Then, for z∈ℂ with Rez>0, the degenerate polygamma function ψλ∗mz, defined by (46), can be expressed as(48)ψλ∗mz=−1mlog1+λλm+1×∫0∞tm1−1+λ−t/λ−11+λ−zt/λdt.
The following recurrence relations for the degenerate polygamma functionψλ∗mz defined by (47) can be obtained from (22)–(24), (35), and (36) as the following.Theorem 9.
Forz∈ℂ,Rez>0,λ∈0,1, and m∈ℕ, the recurrence relations hold true:(49)ψλ∗mz+1=ψλ∗mz+−1mΓm+1zm+1,ψλ∗m1−z=−1mψλ∗mz+−1mπddzmcotπz,ψλ∗mz+n=ψλ∗mz+∑k=0n−1−1mΓm+1z+km+1,ψλ∗m2z=14ψλ∗mz+14ψλ∗mz+12,ψλ∗mnz=1nm+1∑k=1nψλ∗mz+k−1n,Rez>0.
From (25), a series representation of the degenerate polygamma function ψλ∗mz is given in the following result.Theorem 10.
Forz∈ℂ,Rez>0,λ∈0,1, and m∈ℕ, we have(50)ψλ∗mz=−1m+1Γm+1∑n=0∞1z+nm+1.Remark 4.
The degenerate polygamma functionψλ∗mz can be expressed in terms of the generalized zeta function(51)ζm,z=∑n=0∞z+n−mas(52)ψλ∗mz=−1mΓm+1ζm+1,z.
Finally, using (32), a series representation in terms of the Riemann zeta function can be obtained, see the following result.Theorem 11.
Forz∈ℂ,Rez>0,λ∈0,1, and m∈ℕ, we have(53)ψλ∗mz+1=∑n=0∞−1m+n+1Γm+n+1ζm+n+1znn!,m,n∈ℕ.


## 5. Applications

Letz∈ℂ with Rez>0 and n∈ℕ. Then, it can be verified that(54)3F2−n+2,z+1,1z+n+1,2;1=z+nz−n+1×2F1−n+1,zz+1;1−1.Now, we can directly use the integral transform of Gauss hypergeometric function (see [29]) and the formulae:(55)Γ12Γ2z=22z−1ΓzΓz+12,Rez>0,(56)2F1−n+2,zz+1;1=2−zΓz+nΓn−1/2Γz/2+nΓz/2+n−1/2,Rez>0.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:(57)3F2−n+2,z+1,1z+n+1,2;1=z+1z×ψλ∗12+ψλ∗z+1−ψλ∗12z+1−ψλ∗12z+1,Rez>0.Similarly, we can present another identity involving hypergeometric function in terms of our degenerate digamma function in the following form:(58)4F31,1,1,−n2,2,z+1;1×ψλ∗n+2−logλlog1+λψλ∗z+n+1−ψλ∗z−∑s=1nψλ∗s+1−logλ/log1+λz+1,Rez>0.



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