A Note on Bell-Based Bernoulli and Euler Polynomials of Complex Variable

1  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),(1)

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,(2)

and

(2ez+1)αeξz=∑j=0∞Ej(α)(ξ)zjj!,∣z∣<π,(3)

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]),(4)

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]).(5)

The Stirling numbers of the second kind are defined by

ξj=∑r=0jS2(j,r)(ξ)r,(see[12,14,15]).(6)

By (5), we note that

1r!(ez−1)r=∑j=r∞S2(j,r)zjj!,(see[2–22]).(7)

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

eξ(ez−1)=∑j=0∞Belj(ξ)zjj!.(8)

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).(9)

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!.(10)

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!,(11)

zez−1eξzsin⁡ηz=∑j=0∞Bj(ξ+iη)−Bj(ξ−iη)2izjj!=∑j=0∞Bj(s)(ξ,η)zjj!,(12)

and

2ez+1eξzcos⁡ηz=∑j=0∞Ej(ξ+iη)+Ej(ξ−iη)2zjj!=∑j=0∞Ej(c)(ξ,η)zjj!,(13)

2ez+1eξzsin⁡ηz=∑j=0∞Ej(ξ+iη)−Ej(ξ−iη)2izjj!=∑j=0∞Ej(s)(ξ,η)zjj!,(14)

respectively.

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

eξzcos⁡ηz=∑r=0∞Cr(ξ,η)zrr!,(15)

and

eξzsin⁡ηz=∑r=0∞Sr(ξ,η)zrr!,(16)

where

Cr(ξ,η)=∑j=0[r2](−1)j(r2j)ξr−2jη2j,(17)

and

Sr(ξ,η)=∑j=0[r−12](r2j+1)(−1)jξr−2j−1η2j+1.(18)

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.

2  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!.(19)

On the other hand, we suppose that

e(ξ+iη)z=eξzeiηz=eξz(cos⁡ηz+isin⁡ηz).(20)

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),(21)

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).(22)

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!,(23)

and

(zez−1)αeξzsin⁡ηzeζ(ez−1)=∑j=0∞(BelBj(α)(ξ+iη,ζ)−BelBj(α)(ξ−iη;ζ)2i)zjj!.(24)

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!,(25)

and

(zez−1)αeξzsin⁡ηzeζ(ez−1)=∑j=0∞BelBj(α,s)(ξ,η;ζ)zjj!,(26)

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,(27)

BelBj(α,s)(ξ,η;ζ)=BelBj(α)(ξ+iη;ζ)−BelBj(α)(ξ−iη;ζ)2i.(28)

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

(zez−1)αcos⁡ηz=∑j=0∞Bj(α,c)(η)zjj!,(29)

and

(zez−1)αsin⁡ηz=∑j=0∞Bj(α,s)(η)zjj!,(30)

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!,(31)

and

(zez−1)αeξzsin⁡ηz=∑j=0∞Bj(α,s)(ξ;η)zjj!,(32)

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!,(33)

and

(zez−1)αsin⁡ηzeζ(ez−1)=∑j=0∞BelBj(α,s)(η;ζ)zjj!,(34)

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),(35)

and

BelBj(α,s)(η;ζ)=∑v=0[j−12](j2v+1)(−1)vη2v+1BelBj−2v−1(α)(ζ).(36)

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!,(37)

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!.(38)

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),(39)

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).(40)

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),(41)

and

BelBj(α,s)(ξ,η;ζ)=∑u=0j(ju)BelBj−u(α)(ζ)Su(ξ,η).(42)

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),(43)

BelBj(α,s)(ξ,η;ζ)=∑k=0jBk(α,s)(ξ;η)Belj−k(ζ),(44)

BelBj(α,c)(ξ,η;ζ)=∑k=0jBelBk(α,c)(η;ζ)ξj−k,(45)

and

BelBj(α,s)(ξ,η;ζ)=∑k=0jBelBk(α,s)(η;ζ)ξj−k.(46)

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,(47)

and

BelBj(α,s)(ξ+s,η;ζ)=∑u=0j(ju)BelBu(α,s)(ξ,η,ζ)sj−u.(48)

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),(49)

∂∂ηBelBj(α,c)(ξ,η;ζ)=−jBelBj−1(α,s)(ξ,η;ζ),(50)

and

∂∂ξBelBj(α,s)(ξ,η;ζ)=jBelBj−1(α,s)(ξ,η;ζ),(51)

∂∂ηBelBj(α,s)(ξ,η;ζ)=jBelBj−1(α,c)(ξ,η;ζ).(52)

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),(53)

and

∂∂ζBelBj(α,s)(ξ,η;ζ)=BelBj−1(α−1,s)(ξ,η;ζ).(54)

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!.(55)

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(56)

, and

Belj(s)(ξ,η;ζ)=BelBj(s)(ξ+1,η,ζ)−BelBj(s)(ξ,η;ζ)j+1.(57)

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!.(58)

On the other hand, we have

eξzcos⁡ηzeζ(ez−1)=∑j=0∞Belj(c)(ξ,η;ζ)zjj!.(59)

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),(60)

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).(61)

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)!.(62)

On the other hand, we have

eξzcos⁡ηzeζ(ez−1)=∑j=0∞Belj(c)(ξ,η;ζ)zjj!.(63)

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),(64)

and

BelBj(α,s)(ξ,η;ζ)=∑u=0j∑k=0u(ju)BelBj−u(α,s)(η;ζ)(ξ)kS2(u,k).(65)

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!.(66)