CMES CMES CMES Computer Modeling in Engineering & Sciences 1526-1506 1526-1492 Tech Science Press USA 21418 10.32604/cmes.2022.021418 Article A Note on Bell-Based Bernoulli and Euler Polynomials of Complex Variable A Note on Bell-Based Bernoulli and Euler Polynomials of Complex Variable A Note on Bell-Based Bernoulli and Euler Polynomials of Complex Variable AlamN. 1 KhanW. A. 2wkhan1@pmu.edu.sa ObeidatS. 1 MuhiuddinG. 3 DiabN. S. 1 ZaidiH. N. 1 AltalebA. 1 BachiouaL. 1 Department of Basic Sciences, Deanship of Preparatory Year, University of Ha’il, Ha’il, 2440, Saudi Arabia Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al Khobar, 31952, Saudi Arabia Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk, 71491, Saudi Arabia Corresponding Author: W. A. Khan. Email: wkhan1@pmu.edu.sa 27 09 2022 135 1 187 209 13 1 2022 23 5 2022 © 2023 Alam et al. 2023 Alam et al. This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this article, we construct the generating functions for new families of special polynomials including two parametric kinds of Bell-based Bernoulli and Euler polynomials. Some fundamental properties of these functions are given. By using these generating functions and some identities, relations among trigonometric functions and two parametric kinds of Bell-based Bernoulli and Euler polynomials, Stirling numbers are presented. Computational formulae for these polynomials are obtained. Applying a partial derivative operator to these generating functions, some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained. In addition, some remarks and observations on these polynomials are given.

 Bernoulli polynomials euler polynomials bell polynomials stirling numbers Introduction

Special polynomials and numbers possess much importance in multifarious areas of sciences such as physics, mathematics, applied sciences, engineering and other related research fields covering differential equations, number theory, functional analysis, quantum mechanics, mathematical analysis, mathematical physics and so on (see [1–22]) and see also each of the references cited therein. For example, Bernoulli polynomials and numbers are closely related to the Riemann zeta function, which possesses a connection with the distribution of prime numbers. Some of the most significant polynomials in the theory of special polynomials are the Bell, Euler, Bernoulli, Hermite, and Genocchi polynomials. Recently, the aforesaid polynomials and their diverse generalizations have been densely considered and investigated by many physicists and mathematicians (see [1–18,22]) and see also the references cited therein (see [6–9,14–17]). The class of Appell polynomial sequence is one of the significant classes of polynomials sequence [1]. In applied mathematics, theoretical physics, approximation theory, and several other mathematics branches. The set of Appell polynomial sequence is closed under the operation of umbral composition of polynomial sequences. The Appell polynomial sequence can be given by the following generating function:

A(ξ,z)=A(z)eξz=∑j=0∞Aj(ξ)zjj!.

The power series A(z) given by

A(z)=A0+A1z1!+A2z22!+⋯+Ajzjj!+⋯=∑j=0∞Ajzjj!,(A0≠0),where Ai{i=1,2,3,⋯} are real coffiecients. It is easy to see that for any A(z), the derivative of A(z) satisfies

ddzAj(z)=jAj−1(z),j=0,1,2,3,⋯.

The Bell-based Bernoulli and Bell-based Bernoulli polynomials of the first kind are the special cases of Appell polynomials (see [2,18]).

The generalized Bernoulli and Euler polynomials of order α are defined by (see [2–5])

(zez−1)αeξz=∑j=0∞Bj(α)(ξ)zjj!,∣z∣<2π,and

(2ez+1)αeξz=∑j=0∞Ej(α)(ξ)zjj!,∣z∣<π,respectively.

At the point ξ=0, Bj(α)=Bj(α)(0) and Ej(α)=Ej(α)(0) are the Bernoulli and Euler numbers of order α.

For j≥0, Stirling numbers of the first kind are defined by

(ξ)j=∑r=0jS1(j,r)ξr,(see[7–9]),where (ξ)0=1, and (ξ)j=ξ(ξ−1)⋯(ξ−j+1),(ξ≥1). From(4), we see that

1k!(log⁡(1+z))k=∑j=k∞S1(j,k)zjj!,(k≥0),(see[10–17]).

The Stirling numbers of the second kind are defined by

ξj=∑r=0jS2(j,r)(ξ)r,(see[12,14,15]).

By (5), we note that

1r!(ez−1)r=∑j=r∞S2(j,r)zjj!,(see[2–22]).

The generating function of Bell polynomials Belj(ξ) are defined by (see [7])

eξ(ez−1)=∑j=0∞Belj(ξ)zjj!.

In the special case ξ=1, Belj=Belj(1),(j≥0) are the Bell numbers. From (1.7) and (1.8), we have

Belj(ξ)=∑r=0jS2(j,r)ξr,(j≥0).

Recently, Duran et al. [2] introduced the generalized Bell-based Bernoulli polynomials are defined by

(zez−1)αeξz+η(ez−1)=∑j=0∞BelBj(α)(ξ;η)zjj!.

At the point ξ=η=1, BelBj(α)=BelBj(α)(1;1) are the generalized Bell-based Bernoulli numbers.

Kim et al. [13] and Jamei et al. [4,5] introduced the Bernoulli and Euler polynomials of complex variable are defined by

zez−1eξzcos⁡ηz=∑j=0∞Bj(ξ+iη)+Bj(ξ−iη)2zjj!=∑j=0∞Bj(c)(ξ,η)zjj!,

zez−1eξzsin⁡ηz=∑j=0∞Bj(ξ+iη)−Bj(ξ−iη)2izjj!=∑j=0∞Bj(s)(ξ,η)zjj!,and

2ez+1eξzcos⁡ηz=∑j=0∞Ej(ξ+iη)+Ej(ξ−iη)2zjj!=∑j=0∞Ej(c)(ξ,η)zjj!,

2ez+1eξzsin⁡ηz=∑j=0∞Ej(ξ+iη)−Ej(ξ−iη)2izjj!=∑j=0∞Ej(s)(ξ,η)zjj!,respectively.

Also they have prove that (see [4,5,8,9,19,20,21])

eξzcos⁡ηz=∑r=0∞Cr(ξ,η)zrr!,and

eξzsin⁡ηz=∑r=0∞Sr(ξ,η)zrr!,where

Cr(ξ,η)=∑j=0[r2](−1)j(r2j)ξr−2jη2j,and

Sr(ξ,η)=∑j=0[r−12](r2j+1)(−1)jξr−2j−1η2j+1.

Motivated by the importance and potential applications in certain problems in number theory, combinatorics, classical and numerical analysis and physics, several families of Bernoulli and Euler polynomials and special polynomials have been recently studied by many authors, see [8,9,19–21]. Recently, Kim et al. [13,16] have introduced the degenerate Bernoulli and degenerate Euler polynomials of a complex variable. By separating the real and imaginary parts, they introduced the parametric kinds of these degenerate polynomials. The manuscript of this paper is arranged as follows. In Section 2, we introduce parametric kinds of Bell-based Bernoulli polynomials and prove several identities of Bell-based Bernoulli polynomials by using different analytical means and applying generating functions. In Section 3, we establish parametric kinds of Bell-based Euler polynomials and investigate some identities of these polynomials.

 Bell-Based Bernoulli Polynomials of Complex Variable

In this section, we consider the Bell-based Bernoulli polynomials of complex variable and deduce some identities of these polynomials. First, we present the following definition as

(zez−1)αe(ξ+iη)zeζ(ez−1)=∑j=0∞BelBj(α)(ξ+iη;ζ)zjj!.

On the other hand, we suppose that

e(ξ+iη)z=eξzeiηz=eξz(cos⁡ηz+isin⁡ηz).

Thus, by (19) and (20), we have

∑j=0∞BelBj(α)(ξ+iη;ζ)zjj!=(zez−1)αe(ξ+iη)zeζ(ez−1)=(zez−1)αeξz(cos⁡ηz+isin⁡ηz)eζ(ez−1),and

∑j=0∞BelBj(α)(ξ−iη;ζ)zjj!=(zez−1)αe(ξ−iη)zeζ(ez−1)=(zez−1)αeξz(cos⁡ηz−isin⁡ηz)eζ(ez−1).

From (21) and (22), we get

(zez−1)αeξzcos⁡ηzeζ(ez−1)=∑j=0∞(BelBj(α)(ξ+iη;ζ)+BelBj(α)(ξ−iη;ζ)2)zjj!,and

(zez−1)αeξzsin⁡ηzeζ(ez−1)=∑j=0∞(BelBj(α)(ξ+iη,ζ)−BelBj(α)(ξ−iη;ζ)2i)zjj!.

Definition 2.1. Let j≥0. We define two parametric kinds of cosine Bell-based Bernoulli polynomials BelBj(α,c)(ξ,η;ζ) and sine Bell-based Bernoulli polynomials BelBj(α,s)(ξ,η;ζ), for non negative integer j are defined by

(zez−1)αeξzcos⁡ηzeζ(ez−1)=∑j=0∞BelBj(α,c)(ξ,η;ζ)zjj!,and

(zez−1)αeξzsin⁡ηzeζ(ez−1)=∑j=0∞BelBj(α,s)(ξ,η;ζ)zjj!,respectively.

Note that BelBj(α,c)(ξ,0;0)=Bj(α)(ξ),BelBj(α,s)(0,0;0)=0,(j≥0).

From (23)–(26), we have

BelBj(α,c)(ξ,η;ζ)=BelBj(α)(ξ+iη;ζ)+BelBj(α)(ξ−iη;ζ)2,

BelBj(α,s)(ξ,η;ζ)=BelBj(α)(ξ+iη;ζ)−BelBj(α)(ξ−iη;ζ)2i.

Remark 2.1. For ξ=ζ=0 in (25) and (26), we get

(zez−1)αcos⁡ηz=∑j=0∞Bj(α,c)(η)zjj!,and

(zez−1)αsin⁡ηz=∑j=0∞Bj(α,s)(η)zjj!,respectively.

It is clear that

Bj(α,c)(0)=Bj(α,c),Bj(α,s)(0)=0,(j≥0).

Remark 2.2. Letting ζ=0 in (25) and (26), we obtain

(zez−1)αeξzcos⁡ηz=∑j=0∞Bj(α,c)(ξ;η)zjj!,and

(zez−1)αeξzsin⁡ηz=∑j=0∞Bj(α,s)(ξ;η)zjj!,respectively.

Remark 2.3. On setting ξ=0 in (25) and (26), we acquire

(zez−1)αcos⁡ηzeζ(ez−1)=∑j=0∞BelBj(α,c)(η;ζ)zjj!,and

(zez−1)αsin⁡ηzeζ(ez−1)=∑j=0∞BelBj(α,s)(η;ζ)zjj!,respectively.

Now, we start some basic properties of these polynomials.

Theorem 2.1. Let j≥0. Then

BelBj(α,c)(η;ζ)=∑v=0[j2](j2v)(−1)vη2vBelBj−2v(α)(ζ),and

BelBj(α,s)(η;ζ)=∑v=0[j−12](j2v+1)(−1)vη2v+1BelBj−2v−1(α)(ζ).

Proof. By (33) and (34), we can derive the following equations:

∑j=0∞BelBj(α,c)(η;ζ)zjj!=(zez−1)αcos⁡ηzeζ(ez−1)=∑j=0∞BelBj(α,c)(ζ)zjj!∑v=0∞(−1)vη2vzv2v!.=∑j=0∞(∑v=0[j2](j2v)(−1)vη2vBelBj−2v(α)(ζ))zjj!,and

∑j=0∞BelBj(α,s)(η;ζ)zjj!=(zez−1)αsin⁡ηzeζ(ez−1)=∑j=0∞(∑v=0[j−12](j2v+1)(−1)vη2v+1BelBj−2v−1(α)(ζ))zjj!.

Therefore, by (37) and (38), we get (35). Similarly, we can easily obtain (36).

Theorem 2.2. Let j≥0. Then

BelBj(α)(ξ+iη;ζ)=∑u=0j(ju)(ξ+iη)j−uBelBu(α)(ζ)=∑u=0j(ju)(iη)j−uBelBu(α)(ξ;ζ),and

BelBj(α)(ξ−iη;ζ)=∑u=0j(ju)(ξ−iη)j−uBelBu(α)(ζ)=∑u=0j(ju)(−1)j−u(iη)j−uBelBu(α)(ξ;ζ).

Proof. By using (21) and (22), we obtain (39) and (40). So we omit the proof.

Theorem 2.3. Let j≥0. Then

BelBj(α,c)(ξ,η;ζ)=∑u=0j(ju)BelBj−u(α)(ζ)Cu(ξ,η),and

BelBj(α,s)(ξ,η;ζ)=∑u=0j(ju)BelBj−u(α)(ζ)Su(ξ,η).

Proof. Consider

(∑j=0∞ajzjj!)(∑v=0∞bvzvv!)=∑j=0∞(∑v=0jaj−vbv)zjj!.

Now

∑j=0∞BelBj(α,c)(ξ,η,ζ)zjj!=(zez−1)αeξzcos⁡ηzeζ(ez−1)

=(∑j=0∞BelBj(α)(ζ)zjj!)(∑v=0∞Cv(ξ,η)zjj!)

=∑j=0∞(∑v=0j(jv)BelBj−v(α)(ζ)Cv(ξ,η))zjj!,which proves (41). The proof of (42) is similar.

Theorem 2.4. For every j∈Z+, we have

BelBj(α,c)(ξ,η;ζ)=∑k=0jBk(α,c)(ξ;η)Belj−k(ζ),

BelBj(α,s)(ξ,η;ζ)=∑k=0jBk(α,s)(ξ;η)Belj−k(ζ),

BelBj(α,c)(ξ,η;ζ)=∑k=0jBelBk(α,c)(η;ζ)ξj−k,and

BelBj(α,s)(ξ,η;ζ)=∑k=0jBelBk(α,s)(η;ζ)ξj−k.

Proof. Using (25) and (26), we obtain (43)–(46). Here, we omit the proof of the theorem.

Theorem 2.5. Let j≥0. Then

BelBj(α,c)(ξ+s,η;ζ)=∑u=0j(ju)BelBu(α,c)(ξ,η,ζ)sj−u,and

BelBj(α,s)(ξ+s,η;ζ)=∑u=0j(ju)BelBu(α,s)(ξ,η,ζ)sj−u.

Proof. By changing ξ with ξ+s in (25), we have

∑j=0∞BelBj(α,c)(ξ+s,η;ζ)zjj!=(zez−1)αeξzcos⁡ηzeszeζ(ez−1)=(∑j=0∞BelBj(α,c)(ξ,η;ζ)zjj!)(∑u=0∞suzuu!)=∑j=0∞(∑u=0j(ju)BelBu(α,c)(ξ,η;ζ)sj−u)zjj!,which complete the proof (47). The result (48) can be similarly proved.

Theorem 2.6. Let j≥1. Then

∂∂ξBelBj(α,c)(ξ,η;ζ)=jBelBj−1(α,c)(ξ,η;ζ),

∂∂ηBelBj(α,c)(ξ,η;ζ)=−jBelBj−1(α,s)(ξ,η;ζ),and

∂∂ξBelBj(α,s)(ξ,η;ζ)=jBelBj−1(α,s)(ξ,η;ζ),

∂∂ηBelBj(α,s)(ξ,η;ζ)=jBelBj−1(α,c)(ξ,η;ζ).

Proof. Eq. (25) yields

∑j=1∞∂∂ξBelBj(α,c)(ξ,η;ζ)zjj!=(zez+1)αzeξzcos⁡ηzeζ(ez−1)=∑j=0∞BelBj(α,c)(ξ,η;ζ)zj+1j!=∑j=1∞BelBj−1(α,c)(ξ,η;ζ)zj(j−1)!=∑j=1∞jBelBj−1(α,c)(ξ,η;ζ)zjj!,proving (49). Other (50), (51) and (52) can be similarly derived.

Theorem 2.7. Let j≥1. Then

∂∂ζBelBj(α,c)(ξ,η;ζ)=BelBj−1(α−1,c)(ξ,η;ζ),and

∂∂ζBelBj(α,s)(ξ,η;ζ)=BelBj−1(α−1,s)(ξ,η;ζ).

Proof. By (25), we have

∂∂ζ∑j=1∞BelBj(α,c)(ξ,η;ζ)zjj!=∂∂ζ(zez−1)αeξzcos⁡ηzeζ(ez−1)=(zez−1)αeξzcos⁡ηz∂∂ζeζ(ez−1)=(zez−1)αeξzcos⁡ηz(ez−1)eζ(ez−1)=(zez−1)α−1eξzcos⁡ηzeζ(ez−1)z=∑j=1∞BelBj−1(α−1,c)(ξ,η;ζ)zjj!.

The complete proof of (53). The proof of (54) is similar.

Theorem 2.8. For j≥0. Then

Belj(c)(ξ,η;ζ)=BelBj(c)(ξ+1,η;ζ)−BelBj(c)(ξ,η;ζ)j+1, and

Belj(s)(ξ,η;ζ)=BelBj(s)(ξ+1,η,ζ)−BelBj(s)(ξ,η;ζ)j+1.

Proof. By (25), we have

eξzcos⁡ηzeζ(ez−1)=ez−1z∑j=0∞BelBj(c)(ξ,η;ζ)zjj!=1z[∑j=0∞BelBj(c)(ξ+1,η;ζ)zjj!−∑j=0∞BelBj(c)(ξ,η;ζ)zjj!]=∑j=0∞BelBj(c)(ξ+1,η,ζ)−BelBj(c)(ξ,η;ζ)j+1zjj!.

On the other hand, we have

eξzcos⁡ηzeζ(ez−1)=∑j=0∞Belj(c)(ξ,η;ζ)zjj!.

In view of (58) and (59), we get (56). Similarly, we can easily obtain (57).

Theorem 2.9. Let j≥0. Then

Belj(c)(ξ,η;ζ)=j!α!(j+α)!∑m=0j+α∑α=0m(j+αm)S2(m,α)−BelBj+α−m(α,c)(ξ,η;ζ),and

Belj(s)(ξ,η;ζ)=j!α!(j+α)!∑m=0j+α∑α=0m(j+αm)S2(m,α)−BelBj+α−m(α,s)(ξ,η;ζ).

Proof. In definition 2.1, we have

eξzcos⁡ηzeζ(ez−1)=α!(ez−1)αα!∑j=0∞BelBj(α,c)(ξ,η;ζ)zjj!=α!z−α∑α=m∞S2(m,α)zmm!∑j=0∞BelBj(α,c)(ξ,η;ζ)zjj!=α!z−α∑m=0∞∑α=0mS2(m,α)zmm!∑j=0∞BelBj(α,c)(ξ,η;ζ)zjj!=α!∑j=0∞∑m=0j+α∑α=0m(j+αm)S2(m,α)BelBj+α−m(α,c)(ξ,η;ζ)zj(j+α)!.

On the other hand, we have

eξzcos⁡ηzeζ(ez−1)=∑j=0∞Belj(c)(ξ,η;ζ)zjj!.

Therefore, by (62) and (63), we obtain (60). Similarly, we can easily obtain (61).

Theorem 2.10. Let j≥0. Then

BelBj(α,c)(ξ,η;ζ)=∑u=0j∑k=0u(ju)BelBj−u(α,c)(η;ζ)(ξ)kS2(u,k),and

BelBj(α,s)(ξ,η;ζ)=∑u=0j∑k=0u(ju)BelBj−u(α,s)(η;ζ)(ξ)kS2(u,k).

Proof. Using (7) and (25), we find

∑j=0∞BelBj(α,c)(ξ,η;ζ)zjj!=(zez−1)αcos⁡ηzeζ(ez−1)(ez−1+1)ξ=(zez−1)αcos⁡ηzeζ(ez−1)∑k=0∞(ξ)k(ez−1)kk!=(zez−1)αcos⁡ηzeζ(ez−1)∑k=0∞(ξ)k∑u=k∞S2(u,k)zuu!=∑j=0∞∞BellBj(α,c)(η;ζ)zjj!∑u=0∞∑k=0u(ξ)kS2(u,k)zuu!=∑j=0∞∑u=0j∑k=0u(ju)BelBj−u(α,c)(η;ζ)(ξ)kS2(u,k)zjj!.

In view of (25) and (66), we get (64). Similarly, we can easily obtain (65).

 Bell-Based Euler Polynomials of Complex Variable

In this section, we define Bell-based Euler polynomials of complex variable and derive some explicit expressions of these polynomials. Now we start with the following definition as

(2ez+1)αe(ξ+iη)teζ(ez−1)=∑j=0∞BelEj(α)(ξ+iη;ζ)zjj!.

By using (67) and (20), we have

∑j=0∞BelEj(α)(ξ+iη;ζ)zjj!=(2ez+1)αe(ξ+iη)zeζ(ez−1)=(2ez+1)αeξz(cos⁡ηz+isin⁡ηz)eζ(ez−1),and

∑j=0∞BelEj(α)(ξ−iη;ζ)zjj!=(2ez+1)αe(ξ−iη)teζ(ez−1)=(2ez+1)αeξz(cos⁡ηz−isin⁡ηz)eζ(ez−1).

From (68) and (69), we get

(2ez+1)αeξzcos⁡ηzeζ(ez−1)=∑j=0∞(BelEj(α)(ξ+iη;ζ)+BelEj(α)(ξ−iη;ζ)2)zjj!,and

(2ez+1)αeξzsin⁡ηzeζ(ez−1)=∑j=0∞(BelEj(k)(ξ+iη;ζ)−BelEj(k)(ξ−iη;ζ)2i)zjj!.

Definition 3.1. Let j≥0. We define two parametric kinds of cosine Bell-based Euler polynomials BelEj(α,c)(ξ,η;ζ) and sine Bell-based Euler polynomials BelEj(α,s)(ξ,η;ζ), for non negative integer j are defined by

(2ez+1)αeξzcos⁡ηzeζ(ez−1)=∑j=0∞BelEj(α,c)(ξ,η;ζ)zjj!,and

(2ez+1)αeξzsin⁡ηzeζ(ez−1)=∑j=0∞BelEj(α,s)(ξ,η;ζ)zjj!,respectively.

From (70)–(73), we have

BelEj(α,c)(ξ,η;ζ)=BelEj(α)(ξ+iη;ζ)+BelEj(α)(ξ−iη;ζ)2,

BelEj(α,s)(ξ,η;ζ)=BelEj(k)(ξ+iη;ζ)−BelEj(α)(ξ−iη;ζ)2i.

Note that

BelEj(α,c)(ξ,0,0)=Ej(α)(ξ),BelEj(α,s)(ξ,0,0)=0,(j≥0).

Remark 3.1. For ξ=0 in (72) and (73), we get

(2ez+1)αcos⁡ηzeζ(ez−1)=∑j=0∞BelEj(α,c)(η;ζ)zjj!,and

(2ez+1)αsin⁡ηzeζ(ez−1)=∑j=0∞BelEj(α,s)(η;ζ)zjj!,respectively.

Remark 3.2. Letting ζ=0 in (72) and (73), we obtain

(2ez+1)αeξzcos⁡ηz=∑j=0∞Ej(α,c)(η;ζ)zjj!,and

(2ez+1)αeξzsin⁡ηz=∑j=0∞Ej(α,s)(η;ζ)zjj!,respectively.

Remark 3.3. On taking ξ=ζ=0 in (72) and (73), we acquire

(2ez+1)αcos⁡ηz=∑j=0∞Ej(α,c)(η)zjj!,and

(2ez+1)αsin⁡ηz=∑j=0∞Ej(α,s)(η)zjj!,respectively.

Theorem 3.1. Let j≥0. Then

BelEj(α,c)(η;ζ)=∑m=0[j2](j2m)(−1)mη2mBelEj−2m(α)(ζ),and

BelEj(α,s)(η;ζ)=∑m=0[j−12](j2m+1)(−1)mη2m+1BelEj−2m−1(α)(ζ).

Proof. From (78) and (79), we can derive the following equations:

∑j=0∞BelEj(α,c)(η;ζ)zjj!=(2ez+1)αcos⁡ηzeζ(ez−1)=∑j=0∞BelEj(α,c)(ζ)zjj!∑j=0∞(−1)mζ2mzm2m!.=∑j=0∞(∑m=0[j2](j2m)(−1)mη2mBelEj−2m(α)(η))zjj!,and

∑j=0∞BelEj(α,s)(η;ζ)zjj!=(2ez+1)αsin⁡ηzeζ(ez−1)=∑j=0∞(∑m=0[j−12](j2m+1)(−1)mη2m+1BelEj−2m−1(α)(ζ))zjj!.

Therefore, by (82) and (83), we get (80). Similarly, we can easily obtain (81).

Theorem 3.2. Let j≥0. Then

BelEj(α)(ξ+iη;ζ)=∑l=0j(jl)(ξ+iη)j−lBelEl(α)(ζ)=∑l=0j(jl)(iη)j−lBelEl(α)(ξ,ζ),and

BelEj(α)(ξ−iη;ζ)=∑l=0j(jl)(ξ−iη)j−lBelEl(α)(ζ)=∑l=0j(jl)(−1)j−l(iη)j−lBelEl(α)(ξ,ζ).

Proof. By using (68) and (69), we can easily get (84) and (85). So we omit the proof.

Theorem 3.3. Let j≥0. Then

BelEj(α,c)(ξ,η;ζ)=∑m=0j(jm)BelEj−m(α)(ζ)Cm(ξ,η),and

BelEj(α,s)(ξ,η;ζ)=∑m=0j(jm)BelEj−m(α)(ζ)Sm(ξ,η).

Proof. Consider the identity, we have

(∑j=0∞ajzjj!)(∑m=0∞bmzmm!)=∑j=0∞(∑m=0jaj−mbm)zjj!.

Now

∑j=0∞BelEj(α,c)(ξ,η;ζ)zjj!=(2ez+1)αeξzcos⁡ηzeζ(ez−1)=(∑j=0∞BelEj(α)(ζ)zjj!)(∑m=0∞Cm(ξ,η)zmm!)=∑j=0∞(∑m=0j(jm)BelEj−m(α)(ζ)Cm(ξ,η))zjj!,which proves (86). The proof of (87) is similar.

Theorem 3.4. Let j≥0. Then

BelEj(α,c)(ξ,η;ζ)=∑k=0jEk(α,c)(η;ζ)Belj−k(ζ),

BelEj(α,s)(ξ,η;ζ)=∑k=0jEk(α,s)(η;ζ)Belj−k(ζ),

BelEj(α,c)(ξ,η;ζ)=∑k=0jBelEk(α,c)(η;ζ)ξj−k,and

BelEj(α,s)(ξ,η;ζ)=∑k=0jBelEk(α,s)(η;ζ)ξj−k,

Proof. Using (72) and (73), we obtain (88)–(90). Here, we omit the proof of the theorem.

Theorem 3.5. Let j≥0. Then

BelEj(α,c)(ξ+s,η;ζ)=∑m=0j(jm)BelEm(α,c)(ξ,η;ζ)sj−m,and

BelEj(α,s)(ξ+r,η;ζ)=∑m=0j(jm)BelEm(α,s)(ξ,η;ζ)zj−m.

Proof. By changing ξ with ξ+s in (72), we have

∑j=0∞BelEj(α,c)(ξ+s,η;ζ)zjj!=(2ez+1)αeξzcos⁡ηzeszeζ(ez−1)=(∑j=0∞BelEj(α,c)(ξ,η;ζ)zjj!)(∑j=0∞sjzjj!)=∑j=0∞(∑m=0j(jm)BelEm(α,c)(ξ,η;ζ)sj−m)zjj!,which proves (91). The result (92) can be similarly proved.

Theorem 3.6. Let j≥1. Then

∂∂ξBelEj(α,c)(ξ,η;ζ)=jBelEj−1(α,c)(ξ,η;ζ),

∂∂ηBelEj(α,c)(ξ,η;ζ)=−jBelEj−1(α,s)(ξ,η;ζ),and

∂∂ξBelEj(α,s)(ξ,η;ζ)=jBelEj−1(α,s)(ξ,η;ζ),

∂∂ηBelEj(α,s)(ξ,η;ζ)=jBelEj−1(α,c)(ξ,η;ζ).

Proof. Eq. (72) yields

∑j=1∞∂BelEj(α,c)(ξ,η;ζ)∂ξzjj!=(2ez+1)αzeξzcos⁡ηzeζ(ez−1)=∑j=0∞BelEj(α,c)(ξ,η;ζ)zj+1j!=∑j=1∞BelEj−1(α,c)(ξ,η;ζ)zj(j−1)!=∑j=1∞ jBelEj−1(α,c)(ξ,η;ζ)zjj!,proving (93). Other (94)–(96) can be similarly derived.

Theorem 3.7. Let j≥0. Then

∂∂ζBelEj(α,c)(ξ,η;ζ)=BelEj(α,c)(ξ+1,η;ζ)−BelEj(α,c)(ξ,η;ζ),and

∂∂ζBelEj(α,s)(ξ,η;ζ)=BelEj(α,s)(ξ+1,η;ζ)−BelEj(α,s)(ξ,η;ζ).

Proof. By definition (72), we have

∂∂ζ∑j=0∞BelEj(α,c)(ξ,η;ζ)zjj!=∂∂ζ(2ez+1)αeξzcos⁡ηzeζ(ez−1)=(2ez+1)αeξzcos⁡ηz∂∂ζeζ(ez−1)=(2ez+1)αeξzcos⁡ηz(ez−1)eζ(ez−1)=∑j=0∞[BelEj(α,c)(ξ+1,η;ζ)−BelEj(α,c)(ξ,η;ζ)]zjj!.

The complete proof of the result (97). The proof of (98) is similar.

Theorem 3.8. Let j≥0. Then

Belj(c)(ξ,η;ζ)=12[BelEj(c)(ξ+1,η,ζ)+BelEj(c)(ξ,η;ζ)],and

Belj(s)(ξ,η;ζ)=12[BelEj(s)(ξ+1,η,ζ)+BelEj(s)(ξ,η;ζ)].

Proof. Using definition 3.1, we have

eξzcos⁡ηzeζ(ez−1)=ez+12∑j=0∞BelEj(c)(ξ,η;ζ)zjj!=12[∑j=0∞BelEj(c)(ξ+1,η,ζ)zjj!+∑j=0∞BelEj(c)ξ,η;ζ)zjj!]=12∑j=0∞[BelEj(c)(ξ+1,η;ζ)+BelEj(c)ξ,η;ζ)]zjj!.

On the other hand, we have

eξzcos⁡ηzeζ(ez−1)=∑j=0∞Belj(c)(ξ,η;ζ)zjj!.

In view of (102) and (103), we get (100). Similarly, we can easily obtain (101).

Theorem 3.9. Let j≥0. Then

BelEj(α,c)(ξ,η;ζ)=∑l=0j∑k=0l(jl)BelEj−l(α,c)(η;ζ)(ξ)kS2(l,k),and

BelEj(α,s)(ξ,η;ζ)=∑l=0j∑k=0l(jl)BelEj−l(α,s)(η;ζ)(ξ)kS2(l,k).

Proof. Using (7) and (72), we find

∑j=0∞BelEj(α,c)(ξ,η;ζ)zjj!=(2ez+1)αcos⁡ηzeζ(ez−1)(ez−1+1)ξ=(2ez+1)αcos⁡ηzeζ(ez−1)∑k=0∞(ξ)k(ez−1)kk!=(2ez+1)αcos⁡ηzeζ(ez−1)∑k=0∞(ξ)k∑l=k∞S2(l,k)zll!=∑j=0∞BelEj(α,c)(η;ζ)zjj!∑l=0∞∑k=0l(ξ)kS2(l,k)zll!=∑j=0∞∑l=0j∑k=0l(jl)BelEj−l(α,c)(η;ζ)(ξ)kS2(l,k)zjj!.

In view of (72) and (106), we get (104). Similarly, we can easily obtain (105).

 Computational Values and Graphical Representations of Bell-Based Bernoulli Polynomials of Complex Variable

In this section, certain zeros of the Bell-based Bernoulli polynomials of complex variable BelBj(α,c)(ξ,η;ζ) and BelBj(α,s)(ξ,η;ζ) and beautifully graphical representations are shown.

The first few of them are

BelB0(α,c)(ξ,η;ζ)=1,

BelB1(α,c)(ξ,η;ζ)=−α2+ζ+ξ,

BelB2(α,c)(ξ,η;ζ)=α24+(−ζ−ξ−112)α−η2+(ζ+ξ)2+ζ,

BelB3(α,c)(ξ,η;ζ)=ζ−18(α−2(ζ+ξ))(α2−(4ζ+4ξ+1)α+4(ζ+ξ)2+12(ζ−η2)),

BelB4(α,c)(ξ,η;ζ)=ζ4+4ξζ3+6ζ3+6ξ2ζ2+12ξζ2+7ζ2+4ξ3ζ+6ξ2ζ−2αζ+4ξζ−4α(ζ+ξ)ζ+ζ+η4+ξ4+1240α(5α(3(α−2)α+1)+2)−12(α−1)α2(ζ+ξ)+12α(3α−1)((ζ+ξ)2+ζ)−2α(ζ+ξ)((ζ+ξ)2+ζ)+12η2(−3α2+12(ζ+ξ)α+α−12((ζ+ξ)2+ζ)), andBelB0(α,s)(ξ,η;ζ)=0,

BelB1(α,s)(ξ,η;ζ)=η,

BelB2(α,s)(ξ,η;ζ)=2η(−α2+ζ+ξ),

BelB3(α,s)(ξ,η;ζ)=14η(3α2−12α(ζ+ξ)−α+12((ζ+ξ)2+ζ)−4η2),

BelB4(α,s)(ξ,η;ζ)=4ζη−12η(α−2(ζ+ξ))(α2−α(4ζ+4ξ+1)+4(ζ+ξ)2+12ζ−4η2).

Table 1 shows some numerical values of Bell-based Bernoulli polynomials of a complex variable.

Numerical values of Bell-based Bernoulli polynomials of a complex variable <inline-formula id="ieqn-59"><mml:math id="mml-ieqn-59"><mml:msub><mml:mi></mml:mi><mml:mrow><mml:mi>B</mml:mi><mml:mi>e</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mrow><mml:mi mathvariant="double-struck">B</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>ξ</mml:mi><mml:mo>,</mml:mo><mml:mi>η</mml:mi><mml:mo>;</mml:mo><mml:mi>ζ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula id="ieqn-60"><mml:math id="mml-ieqn-60"><mml:msub><mml:mi></mml:mi><mml:mrow><mml:mi>B</mml:mi><mml:mi>e</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mrow><mml:mi mathvariant="double-struck">B</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>ξ</mml:mi><mml:mo>,</mml:mo><mml:mi>η</mml:mi><mml:mo>;</mml:mo><mml:mi>ζ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>
j BelBj(α,c)(5,6;2) BelBj(α,s)(5,6;2)
0 1.0 0
1 6.0 6.0
2 1.83333 72.0
3 −397.0 465.0
4 −5123.9 840.0
5 −38622.3 −21381.0
6 −158258 −341508
7 505053 318760×101
8 1.81135×107 −2.1281×107
9 2.29789×108 −8.71221×108
10 2.19881×109 2.12412×108

Fig. 1 shows the plot for the Bell-based Bernoulli polynomial BelBj(α,c)(ξ,η;ζ) with (ξ,η;ζ)=(5,6;2). Fig. 2 shows the plot for 3D Bell-based Bernoulli polynomials BelBj(α,c)(ξ,η;ζ).

Bell-based Bernoulli polynomials <inline-formula id="ieqn-4056"><mml:math id="mml-ieqn-4056"><mml:msub><mml:mi></mml:mi><mml:mrow><mml:mi>B</mml:mi><mml:mi>e</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mrow><mml:mi mathvariant="double-struck">B</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>ξ</mml:mi><mml:mo>,</mml:mo><mml:mi>η</mml:mi><mml:mo>;</mml:mo><mml:mi>ζ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>3D Bell-based Bernoulli polynomials <inline-formula id="ieqn-4057"><mml:math id="mml-ieqn-4057"><mml:msub><mml:mi></mml:mi><mml:mrow><mml:mi>B</mml:mi><mml:mi>e</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mrow><mml:mi mathvariant="double-struck">B</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>ξ</mml:mi><mml:mo>,</mml:mo><mml:mi>η</mml:mi><mml:mo>;</mml:mo><mml:mi>ζ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> Computational Values and Graphical Representations of Bell-Based Euler Polynomials of Complex Variable

In this section, certain zeros of the Bell-based Bernoulli polynomials of complex variable BelEj(α,c)(ξ,η;ζ) and BelEj(α,s)(ξ,η;ζ) and beautifully graphical representations are shown.

The first few of them are

BelE0(α,c)(ξ,η;ζ)=1,

BelE1(α,c)(ξ,η;ζ)=−α2+ζ+ξ,

BelE2(α,c)(ξ,η;ζ)=α24+ζ−η2+(ζ+ξ)2−14α(1+4ζ+4ξ),

BelE3(α,c)(ξ,η;ζ)=ζ−18(α−2(ζ+ξ))(α2+12(ζ−η2))+4(ζ+ξ)2−α(3+4ζ+4ξ),

BelE4(α,c)(ξ,η;ζ)=−116(α2−2α3(3+4ζ+4ξ))+3α2(1+8ζ2−8η2+16ζ(1+ξ)+8ξ(1+ξ))+16(ζ4+η4−6η2ξ2+ξ4+ξ3(6+4ξ)+ζ(1+2ξ)(1−6η2+2ξ(1+ξ)))+ζ2(7−6η2+6ξ(2+ξ))−2α(−1−12η2+4(4ζ3+3ζ2(5+4ξ))+ζ(7−12η2+6ξ(3+2ξ))+ξ(−12η2+ξ(3+4ξ))), andBelE0(α,s)(ξ,η;ζ)=0,

BelE1(α,s)(ξ,η;ζ)=η,

BelE2(α,s)(ξ,η;ζ)=η(−α+2(ζ+ξ)),

BelE3(α,s)(ξ,η;ζ)=14η(12ζ−4η2+3(α2+4(ζ+ξ)2−α(1+4ζ+4ξ))),

BelE4(α,s)(ξ,η;ζ)=12η(8ζ−(α−2(ζ+ξ))(α2+12ζ−4η2)+4(ζ+ξ)2−α(3+4ζ+4ξ)).

Table 2 shows some numerical values of Numerical values of Bell-based Euler polynomials of complex variable.

Numerical values of Bell-based Euler polynomials of a complex variable <inline-formula id="ieqn-70"><mml:math id="mml-ieqn-70"><mml:msub><mml:mi></mml:mi><mml:mrow><mml:mi>B</mml:mi><mml:mi>e</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mrow><mml:mi mathvariant="double-struck">E</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>ξ</mml:mi><mml:mo>,</mml:mo><mml:mi>η</mml:mi><mml:mo>;</mml:mo><mml:mi>ζ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula id="ieqn-71"><mml:math id="mml-ieqn-71"><mml:msub><mml:mi></mml:mi><mml:mrow><mml:mi>B</mml:mi><mml:mi>e</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mrow><mml:mi mathvariant="double-struck">E</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>ξ</mml:mi><mml:mo>,</mml:mo><mml:mi>η</mml:mi><mml:mo>;</mml:mo><mml:mi>ζ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>
j BelEj(α,c)(5,6;2) BelEj(α,s)(5,6;2)
0 1.0 0
1 6.0 6.0
2 1.5 72.0
3 −403.0 459.0
4 −5127.0 696.0
5 −37282.0 −22914.0
6 −132625 −345096
7 767443 −302880×101
8 1.93872×107 −1.80647×107
9 2.21045×108 −5.04831×107
10 1.91065×109 4.90648×108

Fig. 3 shows the plot for the Bell-based Euler polynomial BelEj(α,c)(ξ,η;ζ) with (ξ,η;ζ)=(5,6;2).

Bell-based Euler polynomials <inline-formula id="ieqn-4057a"><mml:math id="mml-ieqn-4058"><mml:msub><mml:mi></mml:mi><mml:mrow><mml:mi>B</mml:mi><mml:mi>e</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mrow><mml:mi mathvariant="double-struck">B</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>ξ</mml:mi><mml:mo>,</mml:mo><mml:mi>η</mml:mi><mml:mo>;</mml:mo><mml:mi>ζ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> Conclusions

In the present article, we have considered the parametric kinds of Bell-based Bernoulli and Euler polynomials by making use of the exponential as well as trigonometric functions. We have also derived some analytical properties of our newly introduced parametric polynomials by using the series manipulation technique. Furthermore, it is noticed that, if we consider any Appell polynomials of a complex variable (as discussed in the present article), then we can easily define its parametric kinds by separating the complex variable into real and imaginary parts. Consequently, the results of this article may potentially be used in mathematics, mathematical physics and engineering.

The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.

Funding Statement: This research was funded by Research Deanship at the University of Ha’il, Saudi Arabia, through Project No. RG-21 144.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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