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Hindawi Mathematical Problems in Engineering Volume 2020, Article ID 8614841, 9 pages https://doi.org/10.1155/2020/8614841

Research Article Degenerate Analogues of Euler Zeta, , and Polygamma Functions

Fuli He ,1 Ahmed Bakhet ,1,2 Mohamed Akel,3 and Mohamed Abdalla 3,4

1School of and Statistics, Central South University, Changsha 410083, China 2Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt 3Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt 4Department of Mathematics, College of Science, King Khalid University, Abha 61471, Saudi Arabia

Correspondence should be addressed to Mohamed Abdalla; [email protected]

Received 23 February 2020; Accepted 7 April 2020; Published 6 May 2020

Guest Editor: Praveen Agarwal

Copyright © 2020 Fuli He et al. (is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 , and a degenerate . We present several properties, recurrence relations, infinite series, and representations for these functions. Furthermore, we establish identities involving hypergeometric functions in terms of de- generate digamma function.

1. Introduction framework have been widely considered in the literature, for instance, [24–27]. (e , digamma, and polygamma functions have an In this section, we present some basic properties and increasing and recognized role in fractional differential well-known results on a degenerate which equations, mathematical physics, the theory of special we need in this work. In Section 2, we introduce a degenerate functions, statistics, probability theory, and the theory of Euler zeta function and discuss its region of convergence, infinite series. (e reader may refer, for example, to [1–9]. integral representation, and infinite series representation. In (ese functions are directly connected with a variety of Section 3, we define a degenerate digamma function along special functions such as zeta function, Clausen’s function, with its region of convergence and integral representation. and hypergeometric functions. (e evaluations of series We also give certain recurrence relations and formulae involving ζ(s) and related functions satisfied by the degenerate digamma function. In Section 4, have a long history that can be traced back to Christian we define a degenerate polygamma function and describe its Goldbach (1690–1764) and Leonhard Euler (1707–1783) convergence conditions. Some recurrence relations satisfied (see, for details, [10]). (e Euler zeta function and its by the degenerate polygamma function are also given here. generalizations and extensions have been widely studied Finally, in Section 5, the hypergeometric functions are [11–15]. expressed in terms of the degenerate digamma function. ∗ Later on, these functions arise in the study of matrix- In [26], a degenerate gamma function, denoted Γλ , has valued special functions and in the theory of matrix-valued been defined by orthogonal polynomials, see e.., [16–23] and the references ∞ therein. ∗ − t/λ − 1 Γλ (z) � � (1 + λ) t dt, λ ∈ (0, 1), Re(z) > 0. Motivated by this great importance of these functions, 0 their investigations and generalizations to the degenerate (1) 2 Mathematical Problems in Engineering

(e basic results of this function, given in [26], can be 2. Degenerate Euler Zeta Function summarized in the following lemma. (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]) ( , ) z C Lemma 1. Let λ ∈ 0 1 . 1en, for ∈ with ∞ n (z) , ∗(z) (− 1) Re > 0 Γλ satisfies ζ (s, z) � 2 � . (7) E (n + z)s λz n�0 Γ∗(z + 1) � Γ∗(z), λ log(1 + λ) λ An integral representation of ζE(s, z) is given as ∞ λ − 1 s− 1 Γ∗(1) � , ζE(s, z) � Γ (s) � F(− t, z)t dt, (8) λ log(1 + λ) 0 (2) where 2ezt ∞ tn k+1z(z − )···(z − k) F(t, z) � � � E (z) , (9) ∗ λ 1 ∗ + et n n! Γ (z + 1) � Γ (z − k), k ≥ 0, 1 n�0 λ (log(1 + λ))k+1 λ (3) where En(z) is the Euler polynomial of degree n. When z � 0, En � En(0) are Euler numbers (see, [12, 14]). Kim in k+1 [14] obtained that ζn(− n, z) � En(z), n ≥ 0. ∗ λ k! Γ (k + 1) � , k ∈ N. (4) In this section, we consider a degenerate analogue of the λ ( ( + ))k+1 log 1 λ Euler zeta function which is given as ∞ Also, we can easily show that − 1 s− 1 ζE (s, z) � Γ (s) � Fλ(− t, z)t dt, (10) λ 0 ∗ Corollary 1. Let λ ∈ (0, 1). 1en, Γλ (z) satisfies where λ ∈ (0, 1), s, z ∈ C with Re(s) > 0, Re(z) > 0, and z ∞ n ∗ λ 2 zt t Γ (z) �� � Γ(z), z ∈ C, Re(z) > 0, (5) F (t, z) � (1 + λ) /λ � � Eλ(z) . (11) λ log(1 + λ) λ t/λ n n! 1 +(1 + λ) n�0 where Γ(z) is the gamma function. Moreover, for m, n ∈ N, By (9) and (11), it follows that we have n ∗ ∗ ∗ λ λ (m) (n) � B(m, n) (m + n), ( ) E (z) �� � En(z), (12) Γλ Γλ Γλ 6 n ln(1 + λ) B(., .) where is the function. which is the degenerate Euler polynomial of degree n. From (10) and (11), we obtain that

∞ ∞ ∞ − 1 s− 1 − 1 m − (m+z)t/λ s− 1 Γ (s) � Fλ(− t, z)t dt � Γ (s) � 2 � (− 1) (1 + λ) t dt 0 0 m�0

∞ ∞ s− 1 − m − τ � 2Γ 1(s) � (− 1) � (1 + λ) τ/λ dτ (13) (m + z)s m�0 0

Γ∗(s) ∞ 1 � 2 λ � (− 1)m . (s) (m + z)s Γ m�0

s (us, using (10) and (13), we conclude the following (s, z) � (s, z) λ , ( ) ζE ζE �ln(1+λ) � 15 result. λ where ζE(s, z) is the Euler zeta function defined by (7). Theorem 1. For s, z ∈ C with Re(s) > 0, Re(z) > 0, and Furthermore, from (11), it follows λ ∈ (0, 1), the degenerate Euler zeta function ζ (s, z) defined Eλ ∞ ∞ (− 1)m ∞ in (10) has the following infinite series representation: Γ− 1(s) � F (− t, z)ts− 1dt � Γ− 1(s) � Eλ(z) � ts+m− 1dt. λ n m! ∗ ∞ 0 m�0 0 Γλ (s) m 1 ζE (s, z) � 2 � (− 1) s. (14) λ (s) (m + z) (16) Γ m�0 Moreover, in view of (5), we have Hence, we obtain the following results. Mathematical Problems in Engineering 3

Theorem 2. For s, z ∈ C with Re(s) > 0, Re(z) > 0, and ∞ z − 1 (z) � − c + , λ ∈ (0, 1), the degenerate Euler zeta function ζ (s, z), de- ψ � (25) Eλ n�0 (n + 1)(n + z) fined in (10), satisfies ∞ m ∞ where − 1 λ (− 1) s+m− 1 n ζE (s, z) � Γ (s) � En(z) � t dt. (17) 1 λ m! 0 c � lim ⎛⎝� − log n⎞⎠ � − 0.577215 (26) m�0 n⟶∞ k k�1 And for n ∈ N ∪ {}0 , n is the Euler–Mascheroni constant. Hence, substituting (25) 2πi(− 1) λ λ ζE (− n, z) � E � E (z). (18) into (24), one gets the following. λ n!Γ(− n) n n Theorem 4. For z ∈ C, Re(z) > 0, and λ ∈ (0, 1),

Remark 1. Note that ζE (s, z) is an entire function in the λ n 1 λ ∗(z) � − ⎡⎣ n − � ⎤⎦, complex s− plane. ψλ log� � limn⟶∞ log log(1 + λ) j�0 z + j (27) Remark 2.

lim ζE (− n, z) � En(z) � ζE(− n, z). ( ) ∞ λ⟶0 λ 19 ∗ λ 1 ψλ (z) � log� � − c +(z − 1) � , log(1 + λ) n�0 (n + 1)(n + z) (28) 3. Degenerate Digamma Function ∞ (e digamma function, denoted by ψ(z), is the logarithmic ∗ λ 1 ψλ (z + 1) � log� � − c + z � . (29) derivative of the gamma function given by [6, 16, 28]: log(1 + λ) n�1 n(n + z) d Γ′(z) ∗(z) ψ(z) � log Γ(z) � . (20) Next, the degenerate digamma function ψλ defined dz Γ(z) by (21) can be expressed as a series expression in terms of Riemann’s zeta function. Using In this section, we define a degenerate digamma function ∞ m − − − z as follows: (n + z) 1 � n 1 � � � , (30) n m�0 d Γ∗′ (z) ψ∗(z) � log Γ∗(z) � λ , (21) λ λ ∗ equation (29) can be rewritten as dz Γλ (z) λ ∞ ∞ where Γ∗(z) is the degenerate gamma function defined by ψ∗(z + 1) � log� � − c − � � n− (m+1)(− z)m. λ λ ( + ) (1). Now, we are going to obtain certain functional equations log 1 λ n�1 m�1 ∗ involving the degenerate digamma function ψλ (z). Using (31) (2) and (21), it follows that (us, one gets the following. Γ∗′(z + 1) zΓ∗(z) �′ ∗(z + ) � λ � λ ψλ 1 ∗ ∗ Theorem 5. For z ∈ C, Re(z) > 0, and λ ∈ (0, 1), Γλ (z + 1) zΓλ (z) ∞ ∗ λ m ′ ∗ (22) ψλ (z + 1) � log� � − c − � ζ(m + 1)(− z) . Γλ (z) 1 ∗ 1 ( + ) � + � ψ (z) + , log 1 λ m�1 Γ∗(z) z λ z λ (32) Re(z) > 0. Note that these series converge absolutely for |z| < 1. Generally, we have the following. Using the Legendre duplication formula [29] 1 1 Theorem 3. For n ∈ N, z ∈ C, and Re(z) > 0, we have Γ� �Γ(2z) � 22z− 1Γ(z)Γ�z + � (33) 2 2 n− 1 1 ψ∗(z + n) � ψ∗(z) + � . (23) and (5), one can simply find λ λ z + m m�0 1 1 Γ∗� �Γ∗(2z) � 22z− 1Γ∗(z)Γ∗�z + � , (34) λ 2 λ λ λ 2 Furthermore, using relation (5), we find that 1 1 1 ∗( z) � + ∗(z) + ∗�z + �, (z) . λ ψλ 2 log 2 ψλ ψλ Re > 0 ψ∗(z) � ψ(z) + log� �, (24) 2 2 2 λ log(1 + λ) (35) where ψ is the digamma function defined by (20). According Equation (35) can be extended to an arbitrary integral to Batir [28], we have multiplication of z as follows. 4 Mathematical Problems in Engineering

Theorem 6. For z ∈ C, Re(z) > 0, and λ ∈ (0, 1), Now, substituting t � (1 + λ)− s/λ in (37) gives 1 m j − 1 λ log(1 + λ) ψ∗(mz) � log m + � ψ∗�z + �, Re(z) > 0. ψ∗(z) � − c + log� � + λ m λ m λ log(1 + λ) λ j�1

(36) ∞ − 1 × � �(1 + λ)− t/λ − (1 + λ)− zt/λ��1 − (1 + λ)− t/λ� dt. 0 Figures 1–3 illustrate the degenerate digamma function (39) ∗(z) ( , ) ψλ in (24) at different values for λ ∈ 0 1 . Since log(1 + λ) ∞ Remark 3. Its worth to mention here that all plotted z− 1 � � (1 + λ)− zt/λdt (40) functions in the below figures were multiplied by sin x, since λ 0 Fourier space, for the sake of clarify the results to the reader. and by integrating from 1 to n, it follows that Now, we are going to find the integral representations for ∞ n ∗ − zt/λ 1/λ the degenerate digamma function ψλ (z), defined by (21), as log n � � � (1 + λ) · log (1 + λ) dzdt follows. Note that 0 1 1 ∞ 1 ∞ n 1 � �1 − tz− 1 �(1 − t)− 1dt � � � �1 − tz− 1 �tndt � � � d (1 + λ)− zt/λdt (41) t z 0 n�0 0 0 1 ∞ 1 ∞ 1 � � �(1 + λ)− t/λ − (1 + λ)− nt/λ�dt. �(z − 1) � . t (n + )(n + z) 0 n�0 1 (37) Inserting (41) and ∞ − 1 log(1 + λ) − (z+j)t/λ Hence, using (28) and (37), it can be shown that (z + j) � � (1 + λ) dt (42) λ 0 1 ∗ λ z− 1 − 1 ψλ (z) � − c + log� � + � �1 − t �(1 − t) dt. in (27), we get log(1 + λ) 0 (38)

λ ∞ 1 n log(1 + λ) ψ∗(z) � log� � + lim � ⎣⎢⎡�(1 + λ)− t/λ − (1 + λ)− nt/λ� − � (1 + λ)− (z+j)t/λ⎦⎥⎤dt λ ( + ) n⟶∞ t log 1 λ 0 j�0 λ

∞ − 1 λ − t − 1 log(1 + λ) − zt − t � log� � + lim � �(1 + λ) λt − (1 + λ) λ �1 − (1 + λ) λ� �dt (43) log(1 + λ) n⟶∞ 0 λ

∞ − 1 − nt − 1 log(1 + λ) − zt − t − lim � (1 + λ) λ �t − (1 + λ) λ �1 − (1 + λ) λ� �dt. n⟶∞ 0 λ

Since the last limit equals to zero, it follows 4. Degenerate Polygamma Function ∞ ∗ λ 1 − t/λ (e polygamma function of order m is obtained by taking ψλ (z) � log� � + � � (1 + λ) log(1 + λ) 0 t the (m + 1)th derivative of the logarithm of gamma function (cf. [28]). (us, log(1 + λ) − t/λ − 1 − zt/λ dm dm+1 − �1 − (1 + λ) � (1 + λ) �dt. ψ(m)(z) � ψ(z) � log Γ(z), Re(z) > 0. λ dzm dzm+1 (44) (45)

(e following theorem summarizes the above results. In this section, we define the degenerate polygamma function of order m as dm dm+1 Theorem 7. For z ∈ C, Re(z) > 0, and λ ∈ (0, 1), the de- ψ∗ (m)(z) � ψ∗(z) � log Γ∗(z), Re(z) > 0, ∗ λ dzm λ dzm+1 λ generate digamma function ψλ (z), defined by (10), can be expressed as (38), (39) as well as (44). (46) Mathematical Problems in Engineering 5

5 5 7.5

0 5.0 0 5.0

2.5 –5 –5 2.5 5 5 0 0 0 0 –5 –5 –2.5 –2.5 –5 –5 –5.0 0 –5.0 0 –7.5 5 5

(a) (b)

5 7.5 0 5.0

–5 2.5 5 0 0 –5 –2.5 –5 –5.0 0 –7.5 5

(c)

Figure 1: Absolute plots of the degenerate digamma function. (a) λ � 0.1. (b) λ � 0.5. (c) λ � 1.0.

5 5 4 0 0 4

2 2 –5 –5 4 5 2 0 0 0 0 –2 –4 –5 –2 –5 –2 –5

0 0 –4 –4 5 5

(a) (b) Figure 2: Continued. 6 Mathematical Problems in Engineering

5

0 4

2 –5 5 0 0

–5 –2 –5

0 –4

5

(c)

Figure 2: Real-part plots of the degenerate digamma function. (a) λ � 0.1. (b) λ � 0.5. (c) λ � 1.0.

5 5

0 4 0 4

2 2 –5 –5 5 5 0 0 0 0

–5 –2 –5 –2 –5 –5 –4 0 0 –4

5 5

(a) (b) 5

0 4

2 –5 5 0 0

–5 –2 –5

0 –4

5

(c)

Figure 3: Imagery-part plots of the degenerate digamma function. (a) λ � 0.1. (b) λ � 0.5. (c) λ � 1.0.

∗ ∗ (m) where Γλ (z) is the degenerate gamma function defined by Using (44), an integral representation for ψλ (z), ∗ (1) and ψλ (z) is the degenerate digamma function defined given in the next theorem, can be obtained. by (21). By (24), it follows that Theorem 8. Let λ ∈ (0, 1) and m ∈ N. 1en, for z ∈ C with Re(z) > 0, the degenerate polygamma function ψ∗ (m)(z), ∗ (m)(z) � (m)(z), (z) . ( ) λ ψλ ψ Re > 0 47 defined by (46), can be expressed as Mathematical Problems in Engineering 7

m+1 ∞ log(1 + λ) − m ψ∗ (m)(z) �(− 1)m� � ζ(m, z) � � (z + n) (51) λ λ n�0 as ∞ − 1 × � tm�1 − (1 + λ)− t/λ� (1 + λ)− zt/λdt. 0 ∗ (m) m ( ) ψλ (z) � (− 1) Γ(m + 1)ζ(m + 1, z). 52 (48) 1e following recurrence relations for the degenerate Finally, using (32), a series representation in terms of the ∗ (m) Riemann zeta function can be obtained, see the following polygamma function ψλ (z) defined by (47) can be ob- tained from (22)–(24), (35), and (36) as the following. result.

Theorem 11. For z ∈ C, Re(z) > 0, λ ∈ (0, 1), and m ∈ N, Theorem 9. For z ∈ C, Re(z) > 0, λ ∈ (0, 1), and m ∈ N, we have the recurrence relations hold true: ∞ n ∗ (m) m+n+1 z m ψ (z + 1) � � (− 1) Γ(m + n + 1)ζ(m + n + 1) , (− 1) Γ(m + 1) λ n! ψ∗ (m)(z + 1) � ψ∗ (m)(z) + , n�0 λ λ zm+1 m, n ∈ N. m ∗ (m)( − z) �(− )m ∗ (m)(z) +(− )m d ( z), ψλ 1 1 ψλ 1 π�dz � cot π (53)

n− 1 m ∗ (m) ∗ (m) (− 1) Γ(m + 1) 5. Applications ψλ (z + n) � ψλ (z) + � m+1 , k�0 (z + k) Let z ∈ C with Re(z) > 0 and n ∈ N. (en, it can be verified that 1 1 1 ψ∗ (m)(2z) � ψ∗ (m)(z) + ψ∗ (m)�z + �, λ 4 λ 4 λ 2 (− n + 2), z + 1, 1 (− n + 1), z ⎢ z + n ⎢ F ⎣⎢⎡ ; 1⎦⎥⎤ � ×⎛⎝ F ⎣⎢⎡ ; 1⎦⎥⎤ − 1⎞⎠. 3 2 z(− n + 1) 2 1 n z +(n + 1), 2 z + 1 (m) 1 (m) k − 1 ψ∗ (nz) � � ψ∗ �z + �, Re(z) > 0. ( ) λ nm+1 λ n 54 k�1 (49) Now, we can directly use the integral transform of Gauss (see [29]) and the formulae: From (25), a series representation of the degenerate ∗ (m) (z) 1 2z− 1 1 polygamma function ψλ is given in the following result. Γ� �Γ(2z) � 2 Γ(z)Γ�z + �, Re(z) > 0, (55) 2 2 Theorem 10. For z ∈ C, Re(z) > 0, λ ∈ (0, 1), and m ∈ N, (− n + 2), z Γ(z + n)Γ(n − (1/2)) we have 2F � ; 1� � 2− z , 1 z + 1 Γ((z/2) + n)Γ((z/2) +(n − (1/2))) ∞ 1 ∗ (m) m+1 Re(z) > 0. ψλ (z) �(− 1) Γ(m + 1) � m+1. (50) n� (z + n) 0 (56)

Remark 4. (e degenerate polygamma function Using (54) in (56) and L’Hopitalˆ rule for complex ∗ (m) ψλ (z) can be expressed in terms of the generalized zeta numbers with applying equation (24) yields the following function identity in terms of the degenerate digamma function:

(− n + ), z + , 2 1 1 z + 1 1 1 1 F ⎣⎢⎡ ; 1⎦⎥⎤ � ×�ψ∗� � + ψ∗(z + 1) − ψ∗�(z + 1) � − ψ∗�z + 1 ��, Re(z) > 0. (57) 3 2 z λ λ λ λ z +(n + 1), 2 2 2 2 8 Mathematical Problems in Engineering

Similarly, we can present another identity involving hypergeometric function in terms of our degenerate digamma function in the following form:

, , , − n 1 1 1 λ 4F ⎣⎢⎡ ; 1⎦⎥⎤ ×��ψ∗(n + 2) − log� �� ψ∗(z + n + 1) − ψ∗(z)� 3 λ ( + ) λ λ 2, 2, z + 1 log 1 λ (58) n ψ∗(s + 1) − log(λ/log(1 + λ)) − � λ ⎤⎦, Re(z) > 0. z + s�1 1

Data Availability International Journal of Applied and Computational Mathe- matics, vol. 4, no. 4, p. 101, 2018. Data sharing is not applicable to this article as no datasets [5] C. A. Braumann, J.-C. Cortes,´ L. Jodar,´ and L. Villafuerte, “On were generated or analyzed during the current study. the random gamma function: theory and computing,” Journal of Computational and Applied Mathematics, vol. 335, pp. 142–155, 2018. Conflicts of Interest [6] L. Yin, L. Huang, L. Song, and X. K. Dou, “Some monotonicity (e authors declare that they have no conflicts of interest. properties and inequalities for the generalized digamma and polygamma functions,” Journal of Inequalities and Applica- tions, vol. 2018, Article ID 249, 2018. Authors’ Contributions [7] J. Jiang and L. Liu, “Existence of solutions for a sequential fractional differential system with coupled boundary condi- All the authors contributed equally and significantly to tions,” Boundary Value Problems, vol. 2016, Article ID 159, writing this article. All the authors read and approved the 2016. final manuscript. [8] X. Lin and Z. Zhao, “Iterative technique for a third-order differential equation with three-point nonlinear boundary Acknowledgments value conditions,” Electronic Journal of Qualitative 1eory of Differential Equations, vol. 2016, no. 12, pp. 1–10, 2016. (e authors wish to acknowledge the approval and the [9] R. Xu and F. Meng, “Some new weakly singular integral support of this research study from the Deanship of Sci- inequalities and their applications to fractional differential entific Research in King Khalid University, Abha, Saudi equations,” Journal of Inequalities and Applications, vol. 2016, Arabia, under grant R.G.P-79-41. (e first and second au- Article ID 78, 2016. thors wish to acknowledge the approval and the support of [10] H. Srivastava and J. Choi, Zeta and Q-Zeta Functions and this research study from the National Natural Science Associated Series and , Elsevier, Inc., Amsterdam, Foundation of China (11601525). (e authors are thankful to Netherlands, 2012. Professor Hillal Elshehabey for the figures illustrated in this [11] J. Min, “Zeros and special values of Witten zeta functions and paper. Witten L-functions,” Journal of Number 1eory, vol. 134, pp. 240–257, 2014. [12] M. Kim, “Some series involving the Euler zeta function,” References Turkish Journal of Mathematics, vol. 42, no. 3, pp. 1166–1179, 2018. [1] P. Agarwal, F. Qi, M. Chand, and G. Singh, “Some fractional [13] T. Kim, D. V. Dolgy, L.-C. Jang, and H.-I. Kwon, “Some differential equations involving generalized hypergeometric identities of degenerate q -Euler polynomials under the functions,” Journal of Applied Analysis, vol. 25, no. 1, symmetry group of degree n,” Journal of Nonlinear Sciences pp. 37–44, 2019. and Applications, vol. 9, no. 6, pp. 4707–4712, 2016. [2] P. Agarwal, T. Rassias, G. Singh, and S. Jain, “Certain frac- [14] T. Kim, “Euler numbers and polynomials associated with Zeta tional integral and differential formulas involving the ex- tended incomplete generalized hypergeometric functions,” in functions,” Abstract and Applied Analysis, vol. 2008, Article Mathematical Analysis and Applications, pp. 217–272, ID 581582, 11 pages, 2008. Springer, Cham, Switzerland, 2019. [15] D. S. Kim, H. Y. Kim, D. Kim, and T. Kim, “Identities of [3] M. Chand and Z. Hammouch, “Unified fractional integral symmetry for type 2 Bernoulli and euler polynomials,” formulae involving generalized multiindex ,” Symmetry, vol. 11, no. 5, p. 613, 2019. in International Conference on Computational Mathematics [16] R. Dwivedi and V. Sahai, “On certain properties and ex- and Engineering Sciences, pp. 278–290, Springer, Cham, pansions of zeta matrix function, digamma matrix function Switzerland, 2019, https://doi.org/10.1007/978-3-030-39112- and polygamma matrix function,” Quaestiones Mathematicae, 6_22. vol. 43, no. 1, pp. 97–105, 2020. [4] M. Chand, P. Agarwal, and Z. Hammouch, “Certain se- [17] A. Bakhet, Y. Jiao, and F. He, “On the Wright hypergeometric quences involving product of k-Bessel function,” matrix functions and their fractional calculus,” Integral Mathematical Problems in Engineering 9

Transforms and Special Functions, vol. 30, no. 2, pp. 138–156, 2019. [18] F. He, A. Bakhet, M. Hidan, and M. Abdalla, “Two variables shivley’s matrix polynomials,” Symmetry, vol. 11, no. 2, p. 151, 2019. [19] M. Akel, A. Bakhet, and M. Abdalla, “On the matrix version of degenerate hypergeometric function,” Linear Multilinear Algebra, 2020, In press. [20] He. Fuli, A. Bakhet, M. Hidan, and M. Abdalla, “On the extended hypergeometric matrix functions and their appli- cations for the derivatives of the extended Jacobi matrix polynomial,” Mathematical Problems in Engineering, vol. 2020, Article ID 4268361, 8 pages, 2020. [21] A. Bakhet and F. He, “On 2-variables Konhauser matrix polynomials and their fractional integrals,” Mathematics, vol. 8, no. 2, p. 232, 2020. [22] M. Hidan and M. Abdalla, “A note on the Appell hyper- geometric matrix function,” Mathematical Problems in En- gineering, vol. 2020, Article ID 6058987, 6 pages, 2020. [23] M. Abdalla, “Special matrix functions: characteristics, achievements and future directions,” Linear and Multilinear Algebra, vol. 68, no. 1, pp. 1–28, 2020. [24] T. Kim, “Degenerate Euler zeta function,” Russian Journal of Mathematical Physics, vol. 22, no. 4, pp. 469–472, 2015. [25] T. Kim and D. S. Kim, “Degenerate and degenerate gamma function,” Russian Journal of Mathe- matical Physics, vol. 24, no. 2, pp. 241–248, 2017. [26] Y. Kim, B. Kim, L. Jang, and J. Kwon, “A note on modified degenerate gamma and Laplace transformation,” Symmetry, vol. 10, p. 471, 2019. [27] T. Kim and G. Jang, “A note on degenerate gamma function and degenerate Stirling number of the second kind,” Ad- vanced Studies in Contemporary Mathematics, vol. 28, pp. 207–214, 2018. [28] N. Batir, “On some properties of digamma and polygamma Functions,” Journal of Mathematical Analysis and Applica- tions, vol. 328, no. 1, pp. 452–465, 2007. [29] L. Debnath and D. Bhatta, Integral Transform and 1eir Application, CRC Press, Boca Raton, FL, USA, 2015.