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DOI: 10.32604/cmes.2021.016546

ARTICLE

Some Results on Type 2 Degenerate Poly-Fubini Polynomials and Numbers

Ghulam Muhiuddin¹,*, Waseem A. Khan², Abdulghani Muhyi³ and Deena Al-Kadi⁴

1 Department of Mathematics, University of Tabuk, Tabuk, Saudi Arabia
2 Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al Khobar, Saudi Arabia
3 Department of Mathematics, Hajjah University, Hajjah, Yemen
4 Department of Mathematics and Statistic, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
Corresponding Author: Ghulam Muhiuddin. Email: chishtygm@gmail.com
Received: 19 March 2021; Accepted: 27 May 2021

Abstract: In this paper, we introduce type 2 degenerate poly-Fubini polynomials and derive several interesting characteristics and properties. In addition, we define type 2 degenerate unipoly-Fubini polynomials and establish some certain identities. Furthermore, we give some relationships between degenerate unipoly polynomials and special numbers and polynomials. In the last section, certain beautiful zeros and graphical representations of type 2 degenerate poly-Fubini polynomials are shown.

Keywords: Type 2 degenerate poly-Fubini polynomials; modified degenerate polylogarithm function; unipoly functions

1  Introduction

Special functions have gained posses a considerable prominence in various fields of mathematics, physics and applied sciences (see [1–4]) and the references cited therein). Some of the most worthy polynomials in the theory of special polynomials are the Fubini polynomials (see [5–7]), the type 2 poly-Fubini polynomials (see [8]), the degenerate central Fubini polynomials (see [9,10]) and the new type degenerate poly-Bernoulli polynomials (see [11,12]), the degenerate poly-Cauchy polynomials (see [13]), the degenerate poly-Genocchi polynomials (see [14]). Recently, the aforementioned special polynomials and their various extensions have been widely investigated by many researchers (see [1–32]) and see also each of the references cited therein.

The generating functions of the classical Bernoulli Bm(u) , Euler Em(u) and Genocchi Gm(u) polynomials are given as

ξeξ−1euξ=∑m=0∞Bm(u)ξmm!,|ξ|<2π,2eξ+1euξ=∑m=0∞Em(u)ξmm!,|ξ|<πand2ξeξ+1euξ=∑m=0∞Gm(u)ξmm!,|ξ|<π,(see [18-20]), (1)

respectively.

Kargin [29] defined the 2-variable Fubini polynomials by the following generating function:

euξ1−v(eξ−1)=∑m=0∞Fm(u,v)ξmm!. (2)

Taking u = 0 in Eq. (2), the 2-variable Fubini polynomials Fm(u, v) reduce to the classical Fubini polynomials given by

11−v(eξ−1)=∑m=0∞Fm(v)ξmm!. (3)

We easily can write

Fm(u,−12)=Em(u),Fm(−12)=Em (4)

and

Fm(v)=∑α=0mS2(m,α)α!vα.

For v = 1 in (3), we get the known Fubini numbers Fm(1) = Fm as follows:

12−eξ=∑m=0∞Fmξmm!,(see [5,6,10]). (5)

Some of the applications of Fubini polynomials and numbers can be found in [7–9,20,26,27].

The degenerate form of the exponential function is given as (see [11,12,14,22–29])

eβu(ξ)=(1+βt)uβ,eβ(ξ)eβ1(ξ)=(1+βξ)1β,β∈R. (6)

The function eβu(ξ) is defined by the series

eβu(ξ)=∑m=0∞(u)m,βξmm!, (7)

where the degenerate Pochhammer symbol (u)m, β is specified by

(u)m,β=u(u−β)(u−2β)⋯(u−(m−1)β),(u)0,β=1,(m≥1).

Carlitz et al. [15,16] presented the degenerate Bernoulli polynomials as

ξeβ(ξ)−1eβu(ξ)=ξ(1+βξ)1β−1(1+βξ)uβ=∑m=0∞Bm,β(u)ξmm!, (8)

where Bm,β(0):=Bm,β denotes degenerate Bernoulli numbers.

For α∈Z , the modified degenerate polyexponential function [28] is defined by Kim–Kim to be

Eiα,β(u)=∑m=1∞(1)m,βum(m−1)!mα,(|u∣<1). (9)

Note that

Ei1,β(u)=∑m=1∞(1)m,βumm!=eβ(u)−1. (10)

Kim et al. [14] presented the generating function of the degenerate poly-Genocchi polynomials as

Eiα,β(logβ⁡(1+ξ))eβ(ξ)+1eβu(ξ)=∑m=0∞Gm,β(α)(u)ξmm!,(α∈Z), (11)

where Gm,β(α)(0):=Gm,β(α) denotes the degenerate poly-Genocchi numbers.

The 2-variable degenerate Fubini polynomials Fm, β(u; v) [26] are defined by

11−v((1+βξ)1β−1)(1+βξ)uβ

=∑m=0∞Fm,β(u;v)ξmm!,(see [25,27]), (12)

where Fm, β(0; 1) := Fm, β denotes the degenerate Fubini numbers.

The degenerate Daehee polynomials Dm, β(u) [11] are specified by

logβ⁡(1+ξ)ξ(1+ξ)u=∑m=0∞Dm,β(u)ξmm!,(see [12,20]), (13)

where Dm, β(0) := Dm, β denotes the degenerate Daehee numbers.

The degenerate form of the first kind Stirling numbers are specified by

1α!(logβ⁡(1+ξ))α=∑m=α∞S1,β(m,α)ξmm!,(α≥0),(see [12]). (14)

Note here that limβ→0S1,β(m,α)=S1(m,α) , where S1(m,α) are the first kind Stirling numbers given by

1α!(log⁡(1+ξ))α=∑m=α∞S1(m,α)ξmm!,(α≥0). (15)

The degenerate form of the second kind Stirling numbers are specified by

1α!(eβ(ξ)−1)α=∑m=α∞S2,β(m,α)ξmm!,(see [23]). (16)

Observe here that limβ→0S2,β(m,α)=S2(m,α) , where S2(m, α) are the second kind Stirling numbers given by

1α!(eξ−1)α=∑m=α∞S2(m,α)ξmm!,(see [1-32]). (17)

The following paper is as follows. In Section 2, we define type 2 degenerate poly-Fubini polynomials via the modified degenerate polyexponential functions and obtain certain new properties related to these numbers and polynomials. In Section 3, we consider the type 2 degenerate unipoly-Fubini polynomials and discuss some identities of them. In Section 4, we find some values of type 2 poly-Fubini polynomials and draw some beautiful graphs.

2  Type 2 Degenerate Poly-Fubini Polynomials and Numbers

In the present section, we define type 2 degenerate Fubini polynomials by utilizing the modified degenerate polyexponential function and we derive some interesting relations and formulas related to these polynomials and numbers. We start the following definition as follows.

Let β∈C and α∈Z , we consider the type 2 degenerate poly-Fubini polynomials which are defined by

Eiα,β(logβ⁡(1+ξ))ξ(1−v((1+βξ)1β−1))(1+βξ)uβ=∑m=0∞Fm,β(α)(u;v)ξmm!, (18)

where Fm,β(α)(0;1):=Fm,β(α) denotes the type 2 degenerate poly-Fubini numbers, and logβ⁡(ξ)=1β(ξβ−1) is the compositional inverse of eβ(ξ) satisfying the following relation

logβ⁡eβ(ξ)=eβ(logβ⁡(ξ))=ξ.

For α = 1 in Eq. (18), we get

11−v((1+βξ)1β−1)(1+βξ)uβ=∑m=0∞Fm,β(u;v)ξmm!, (19)

where Fm, β(u; v) denotes the degenerate Fubini polynomials (see Eq. (12)).

Obviously

limβ→0(Eiα,β(logβ⁡(1+ξ))ξ(1−v((1+βξ)1β−1))(1+βξ)uβ)=∑m=0∞limβ→0Fm,β(α)(u;v)ξmm!

=Eiα(log⁡(1+ξ))ξ(1−v(eξ−1))euξ=∑m=0∞Fm(α)(u;v)ξmm!. (20)

Thus, by (18) and (20), we have

limβ→0Fm,β(α)(u;v)=Fm(α)(u;v),(m≥0) (21)

where Fm(α)(u;v) denotes the type 2 poly-Fubini polynomials (see [28]).

Theorem 2.1. For m ≥ 0. Then, we have

Fm,β(α)(v)=∑l=0m∑ρ=0l(ml)Fm−l,β(v)(1)β,ρ+1S1,β(l+1,ρ+1)l+1(ρ+1)α−1. (22)

Proof. By using Eqs. (9), (14) and (18), we observe that

∑m=0∞Fm,β(α)(v)ξmm!=Eiα,β(logβ⁡(1+ξ))ξ(1−v((1+βξ)1β−1))

=11−v((1+βξ)1β−1)1ξ∑ρ=1∞(1)ρ,β(logβ⁡(1+ξ))ρ(ρ−1)!ρα

=11−v((1+βξ)1β−1)1ξ∑ρ=0∞(1)ρ+1,β(logβ⁡(1+ξ))ρ+1(ρ+1)!(ρ+1)α−1

=11−v((1+βξ)1β−1)1ξ∑ρ=0∞(1)ρ+1,β(ρ+1)α−1∑l=ρ+1∞S1,β(l,ρ+1)ξll!

=11−v((1+βξ)1β−1)∑ρ=0∞(1)ρ+1,β(ρ+1)α−1∑l=ρ∞S1,β(l+1,ρ+1)ξl(l+1)!

=(∑s=0∞Fm,β(v)ξmm!)(∑l=0∞∑ρ=0l(1)ρ+1,β(ρ+1)α−1S1,β(l+1,ρ+1)l+1ξll!)

L.H.S=∑m=0∞(∑l=0m∑ρ=0l(ml)Fm−l,β(v)(1)ρ+1,βS1,β(l+1,ρ+1)l+1(ρ+1)α−1)ξmm!. (23)

Therefore, by (23), we obtain the result.

Corollary 2.1. For m ≥ 0. Then, we have

Fm,β(1)(v)=∑l=0m∑ρ=0l(ml)Fm−l,β(v)(1)β,ρ+1S1,β(l+1,ρ+1)l+1. (24)

Theorem 2.2. Let m ≥ 0. Then, we have

Fm,β(α)(u;v)=∑ρ=0m(mρ)Fm−ρ,β(α)(v)(u)ρ,β. (25)

Proof. From (18), we observe that

∑m=0∞Fm,β(α)(u;v)ξmm!=Eiα,β(logβ⁡(1+ξ))ξ(1−v((1+βξ)1β−1))(1+βξ)uβ

=(∑m=0∞Fm,β(α)(v)ξmm!)(∑ρ=0∞(u)ρ,βξρρ!)

=(∑m=0∞Fm,β(α)(v)ξmm!)(∑ρ=0∞(u)ρ,βξρρ!)

L.H.S=∑m=0∞(∑ρ=0m(mρ)Fm−ρ,β(α)(v)(u)ρ,β)ξmm!. (26)

By comparing the coefficients on both sides of Eq. (26), we reach at the following result (25).

In [12], it is well known that the degenerate second kind Bernoulli polynomials are defined by

ξlogβ⁡(1+ξ)(1+ξ)u=∑m=0∞bm,β(u)ξmm!, (27)

where bm, β(0):= bm, β denotes the degenerate second kind Bernoulli numbers.

Theorem 2.3. For m ≥ 0, we have

Fm,β(α)(v)=12∑ρ=0m(mρ)∑ρ1+⋯+ρα−1=ρ(ρρ1+⋯+ρα−1)

×bρ1,β(β−1)ρ1+1bρ2,β(β−1)ρ1+ρ2+1⋯bρα−1,β(β−1)ρ1+⋯+ρα−1+1Fm−ρ,β(v). (28)

Proof. From (9), it follows that

dduEiα,β(logβ⁡(1+u))=ddu∑m=1∞(1)m,β(logβ⁡(1+u))m(m−1)!mα

=(1+u)β−1logβ⁡(1+u)∑m=1∞(1)m,β(logβ⁡(1+u))m(m−1)!mα−1=(1+u)β−1logβ⁡(1+u)Eiα−1,β(logβ⁡(1+u)). (29)

Thus, from (18) and (29), we have

∑m=0∞Fm,β(α)(v)umm!=1u(1−v((1+βu)1β−1))Eiα,β(logβ⁡(1+u))

=1u(1−v((1+βu)1β−1))∫0u(1+ξ)β−1logβ⁡(1+ξ)∫0ξ⋯(1+ξ)β−1logβ⁡(1+ξ)∫0ξ⏟(α−2)−times(1+ξ)β−1logβ⁡(1+ξ)tdt⋯dt

=1u(1−v((1+βu)1β−1))∑ρ=0∞∑ρ1+⋯+ρα−1=ρ(ρρ1+⋯+ρα−1)

×bρ1,β(β−1)ρ1+1bρ2,β(β−1)ρ1+ρ2+1⋯bρα−1,β(β−1)ρ1+⋯+ρα−1+1uρρ!

∑m=0∞Fm,β(α)(v)umm!=12∑m=0∞∑ρ=0m(mρ)∑ρ1+⋯+ρα−1=ρ(ρρ1+⋯+ρα−1)

×bρ1,β(β−1)ρ1+1bρ2,β(β−1)ρ1+ρ2+1⋯bρα−1,β(β−1)ρ1+⋯+ρα−1+1Fm−ρ,β(v)umm!. (30)

Therefore, by (30), we obtain the result.

Corollary 2.2. For α ≥ 2, we have

Fm,β(2)(v)=12∑ρ=0m(mρ)bρ,β(β−1)ρ+1Fm−ρ,β(v).

Theorem 2.4. Let α ≥ 1 and α∈N∪{0} , s∈C , we have

ηα,β(−ρ)=(−1)ρFρ,β(α)(v).

Proof. Let α ≥ 1, be an integer. For s∈C , we define the function ηα, β(s) as

ηα,β(s)=1Γ(s)∫0∞ξs−1ξ(1−v((1+βξ)1β−1))Eiα,β(logβ⁡(1+ξ))dt

=1Γ(s)∫01ξs−1ξ(1−v((1+βξ)1β−1))Eiα,β(logβ⁡(1+ξ))dt

+1Γ(s)∫1∞ξs−1ξ(1−v((1+βξ)1β−1))Eiα,β(logβ⁡(1+ξ))dt. (31)

Here, we note that the second integral converges absolutely for any s∈C , then the second term on the r.h.s. vanishes at non-positive integers. Hence,

lims→−ρ|1Γ(s)∫1∞ξs−1ξ(1−v((1+βξ)1β−1))Eiα,β(logβ⁡(1+ξ))dt|≤1Γ(−ρ)M=0. (32)

Also, for ℜ (s) > 0, the first integral in (32) can be written as

1Γ(s)∑l=0∞Fl,β(α)(v)l!1s+l.

Using (31) and (32), we see that

ηα,β(−ρ)=lims→−ρ1Γ(s)∫01ξs−1ξ(1−v((1+βξ)1β−1))Eiα,β(logβ⁡(1+ξ))dt

=lims→−ρ1Γ(s)∫01ξs−1∑l=0∞Fl,β(α)(v)ξll!dξ=lims→−ρ1Γ(s)∑l=0∞Fl,β(α)(v)s+l1l!

=⋯+0+⋯+0+lims→−ρ1Γ(s)1s+ρFρ,β(α)(v)ρ!+0+0+⋯

=lims→−ρ(Γ(1−s)sin⁡πsπ)s+ρFρ,β(α)(v)ρ!=Γ(1+ρ)cos⁡(πρ)Fρ,β(α)(v)ρ!

=(−1)ρFρ,β(α)(v). (33)

Therefore, by (33), the result is obtained.

Theorem 2.5. Let α ≥ 1 and ρ∈N∪{0} , s∈C , we have

Fm,β(α)(v)=11+v[v∑ρ=0m(mρ)(1)ρ,βFm−ρ,β(α)(v)+∑ρ=0m1(ρ+1)α−1(1)ρ+1,βS1,β(m+1,ρ+1)m+1].

Proof. From (18), we note that

Eiα,β(logβ⁡(1+ξ))ξ=(1−v((1+βξ)1β−1))∑m=0∞Fm,β(α)(v)ξmm!

=∑m=0∞Fm,β(α)(v)ξmm!−y∑m=0∞∑ρ=0m(mρ)(1)ρ,βFm−ρ,β(α)(v)ξmm!+y∑m=0∞Fm,β(α)(v)ξmm!

=(1+v)∑m=0∞Fm,β(α)(v)ξmm!−y∑m=0∞∑ρ=0m(mρ)(1)ρ,βFm−ρ,β(α)(v)ξmm!. (34)

On the other hand,

Eiα,β(logβ⁡(1+ξ))ξ=1ξ∑ρ=1∞(1)ρ,β(logβ⁡(1+ξ))ρ(ρ−1)!ρα

=1ξ∑ρ=0∞(1)ρ+1,β(logβ⁡(1+ξ))ρ+1ρ!(ρ+1)α(ρ+1)!(ρ+1)!

=1ξ∑ρ=0∞(1)ρ+1,β(ρ+1)α−1∑m=ρ+1∞S1,β(m,ρ+1)ξmm!

=∑m=0∞(∑ρ=0m1(ρ+1)α−1(1)ρ+1,βS1,β(m+1,ρ+1)m+1)ξmm!. (35)

Therefore, by (34) and (35), we reach at the desired result.

For α = 1 in Theorem 2.5., the following corollary is obtained.

Corollary 2.3. For ρ∈N∪{0} , we have

Fm,β(v)=11+v[v∑ρ=0m(mρ)(1)ρ,βFm−ρ,β(v)+∑ρ=0m(1)ρ+1,βS1,β(m+1,ρ+1)m+1].

Theorem 2.6. Let m ≥ 0. Then, we have

vFm,β(α)(u+1;v)=(v+1)Fm,β(α)(u;v)−∑l=0m∑ρ=0l(ml)1(ρ+1)α−1S1,β(l+1,ρ+1)l+1(u)m−l,β.

Proof. From (18), we note that

∑m=0∞(Fm,β(α)(u+1;v)−Fm,β(α)(u;v))ξmm!

=Eiα,β(logβ⁡(1+ξ))ξ(1−v((1+βξ)1β−1))(1+βξ)uβ((1+βξ)1β−1)

=Eiα,β(logβ⁡(1+ξ))ξ(1v(((1+βξ)uβ1−v((1+βξ)1β−1))−(1+βξ)uβ))

=1v(Eiα,β(logβ⁡(1+ξ))(1+βξ)uβξ(1−v((1+βξ)1β−1))−Eiα,β(logβ⁡(1+ξ))(1+βξ)uβξ)

=1v(∑m=0∞Fm,β(α)(u;v)ξmm!−∑l=0∞∑ρ=0l1(ρ+1)α−1S1,β(l+1,ρ+1)l+1ξll!∑m=0∞(u)m,βξmm!)

=1v∑m=0∞(Fm,β(α)(u;v)−∑l=0m∑ρ=0l(ml)1(ρ+1)α−1S1,β(l+1,ρ+1)l+1(u)m−l,β)ξmm!. (36)

Upon comparing the coefficients of ξmm! of the above equation, we get the result.

When u = 0 and u = −1 in Theorem (2.6), we get

vFm,β(α)(1;v)=(v+1)Fm,β(α)(v)−∑l=0m∑v=0l(ml)1(ρ+1)α−1S1,β(l+1,ρ+1)l+1,(m≥0).

and

vFm,β(α)(v)=(v+1)Fm,β(α)(−1;v)−∑l=0m∑ρ=0l(ml)1(ρ+1)α−1S1,β(l+1,ρ+1)l+1(−1)m−l,β,(m≥0).

Theorem 2.7. Let m ≥ 0. Then, we have

∑ρ=0m(mρ)Fm−ρ,β(α)(u1;v1)Fρ,β(α)(u2;v2)

=v2Fm,β(α)(u1+u2;v2)−v1Fm,β(α)(u1+u2;v1)v2−v1.

Proof. Now, we observe that

(Eiα,β(logβ⁡(1+ξ))ξ(1−v1((1+βξ)1β−1))(1+βξ)u1β)(Eiα,β(logβ⁡(1+ξ))t(1−v2((1+βξ)1β−1))(1+βξ)u2β)

=Eiα,β(logβ⁡(1+ξ))ξ(v2v2−v1(1+βξ)u1+u2β1−v2((1+βξ)1β−1)−v1v2−v1(1+βξ)u1+u2β1−v2((1+βξ)1β−1))

=∑m=0∞(v2Fm,β(α)(u1+u2;v2)−v1Fm,β(α)(u1+u2;v1)v2−v1)ξmm!. (37)

On the other hand,

(Eiα,β(logβ⁡(1+ξ))ξ(1−v1((1+βξ)1β−1))(1+βξ)u1β)(Eiα,β(logβ⁡(1+ξ))t(1−v2((1+βξ)1β−1))(1+βξ)u2β)

=(∑m=0∞Fm,β(α)(u1;v1)ξmm!)(∑ρ=0∞Fρ,β(α)(u2;v2)ξρρ!)

=∑m=0∞(∑ρ=0m(mρ)Fm−ρ,β(α)(u1;v1)Fρ,β(α)(u2;v2))ξmm!. (38)

Therefore, by (37) and (38), we get the result.

For u1 = u2 = 0 in Theorem 2.7, we have

Corollary 2.4. Let α∈Z and m ≥ 0, we have

∑ρ=0m(mρ)Fm−ρ,β(α)(v1)Fρ,β(α)(v2)=v2Fm,β(α)(v2)−v1Fm,β(α)(v1)v2−v1.

Theorem 2.8. Let m ≥ 0. Then, we have

Fm,β(α)(u;v−1)=(−1)mFm,−β(α)(1−u;−v).

Proof. From (18), we note that

∑m=0∞Fm,β(α)(u;v−1)ξmm!=Eiα,β(logβ⁡(1+ξ))ξ(1−(v−1)((1+βξ)1β−1))(1+βξ)uβ

=Eiα,β(logβ⁡(1+ξ))(1+βξ)uβξ(1−v((1+βξ)1β−1)+(1+βξ)1β−1)

=Eiα,β(logβ⁡(1+ξ))(1+βξ)u−1βξ(1−v(1−βξ)1β)

=Eiα,β(logβ⁡(1+ξ))(1−β(−ξ))−1−uβξ(1+v((1−β(−ξ))−1β−1))

=∑m=0∞Fm,−β(α)(1−u;−v)(−1)mξmm!. (39)

Comparing the coefficients of ξmm! of the above equation, we obtain the result.

Adjusting u = 0 in Theorem 2.8, we get

Corollary 2.5. Let α∈Z and m ≥ 0, we have

Fm,β(α)(v−1)=(−1)mFm,−β(α)(1;−v).

Theorem 2.9. Let α∈Z and m ≥ 0, we have

Fm,β(α)(v)=∑l=0m∑r=0m−l∑ρ=0l(ml)1(ρ+1)α−1(1)ρ+1,βS1,β(l+1,ρ+1)l+1vrS2,β(m−l,r).

Proof. From (18), we see that

∑m=0∞Fm,β(α)(v)ξmm!=Eiα,β(logβ⁡(1+ξ))ξ(1−v((1+βξ)1β−1))

=(Eiα,β(logβ⁡(1+ξ))ξ)(∑r=0∞vr((1+βξ)1β−1)r)

=1ξ(∑ρ=1∞(1)ρ,β(logβ⁡(1+ξ))ρ(ρ−1)!ρα)(∑m=0∞∑r=0mvrS2,β(m,r)ξmm!)

(∑l=0∞∑ρ=0∞1(ρ+1)α−1(1)ρ+1,βS1,β(l+1,ρ+1)l+1ξll!)(∑m=0∞∑r=0mvrS2,β(m,r)ξmm!)

=∑m=0∞(∑l=0m∑r=0m−l∑ρ=0l(ml)1(ρ+1)α−1(1)ρ+1,βS1,β(l+1,ρ+1)l+1vrS2,β(m−l,r))ξmm!. (40)

Therefore, utilizing Eqs. (40), the result is obtained.

Corollary 2.6. Let α∈Z and m ≥ 0, we have

Fm,β(v)=∑l=0m∑r=0m−l∑ρ=0l(ml)(1)ρ+1,βS1,β(l+1,ρ+1)l+1vrS2,β(m−l,r).

In the next section, the type 2 degenerate unipoly-Fubini polynomials are established and, certain explicit expressions and properties are obtained.

3  Type 2 Degenerate Unipoly-Fubini Polynomials and Numbers

Recently, Kim et al. [22] defined the unipoly function by

uk(u∣p)=∑m=1∞p(m)mαum,(α∈Z), (41)

where p denotes any real or complex valued arithmetic function which is a defined on N (set of positive integers).

Moreover,

uk(u∣1)=∑m=1∞ummα=Liα(u),(see [17]), (42)

denotes the ordinary polylogarithm function.

Dolgy et al. [13] introduced the degenerate unipoly function attached to polynomials p (u) as follows:

uα,β(u∣p)=∑i=1∞p(i)(1)i,βuiiα. (43)

We note that

uα,β(u∣1Γ)=Eiα,β(u) (44)

is the modified degenerate polyexponential function.

By using (43), we introduce the degenerate unipoly-Fubini polynomials as follows:

uα,β(logβ⁡(1+ξ)|p)ξ(1−v((1+βξ)1β−1)(1+βξ)uβ=∑m=0∞Fm,β,p(α)(u;v)ξmm!. (45)

In the case when u = 0 and y = 1, Fm,β,p(α)=Fm,β,p(α)(0;1) are called the type 2 degenerate unipoly-Fubini numbers.

Theorem 3.1. Let m ≥ 0. Then, we have

Fm,β,1Γ(α)(u;v)=Fm,β(α)(u;v),(k∈Z).

Proof. Let us take p(m)=1Γβ . Then we have

∑m=0∞Fm,β,1Γ(α)(u;v)ξmm!=uα,β(logβ⁡(1+ξ)|1Γp)ξ(1−v((1+βξ)1β−1)(1+βξ)uβ

=1ξ(1−v((1+βξ)1β−1)∑ρ=1∞(1)ρ,β(logβ⁡(1+ξ))ρρα(ρ−1)!(1+βξ)uβ

=Eiα,β(logβ⁡(1+ξ))ξ(1−v((1+βξ)1β−1)(1+βξ)uβ

=∑m=0∞Fm,β(α)(u;v)ξmm!. (46)

Thus, by (46), the result is obtained.

Theorem 3.2. For m∈N and α∈Z , we have

Fm,β,p(α)(v)=∑l=0∞∑ρ=0l(ml)p(ρ+1)(1)ρ+1,β(ρ+1)!S1,β(ρ+1,l+1)Fm−l,β(v)(ρ+1)α(l+1). (47)

Proof. From (45), we get

∑m=0∞Fm,β,p(α)(v)ξmm!=uα,β(logβ⁡(1+ξ)|p)ξ(1−v((1+βξ)1β−1)

=1ξ(1−v((1+βξ)1β−1)∑ρ=1∞p(ρ)(1)ρ,βρα(logβ⁡(1+ξ))ρ

=1ξ(1−v((1+βξ)1β−1)∑ρ=0∞p(ρ+1)(1)ρ+1,β(ρ+1)!(ρ+1)α∑l=ρ+1∞S1,β(ρ+1,l)ξll!

=(∑j=0∞Fj,β(v)ξjj!)(∑ρ=0∞∑l=0ρp(ρ+1)(1)ρ+1,β(ρ+1)!(ρ+1)αS1,β(ρ+1,l+1)l+1ξll!)

=∑m=0∞(∑l=0∞∑ρ=0l(ml)p(ρ+1)(1)ρ+1,β(ρ+1)!S1,β(ρ+1,l+1)Fm−l,β(v)(ρ+1)α(l+1))ξmm!. (48)

Thus, we complete the proof of this theorem.

In particular,

Fm,β,1Γ(α)(v)=Fm,β(α)(v)=∑l=0∞∑ρ=0l(ml)S1,β(ρ+1,l+1)Fm−l,β(v)(ρ+1)α−1(l+1). (49)

Theorem 3.3. Let m ≥ 0 and α∈Z . Then we have

Fm,β,p(α)(u;v)=∑l=0m∑ρ=0l(ml)Fm−l,β,p(α)(v)(u)(ρ)S2,β(l,ρ;−ρ). (50)

Proof. From (45), we observe that

∑m=0∞Fm,β(α,p)(u;v)ξmm!=uα,β(logβ⁡(1+ξ)|p)ξ(1−v((1+βξ)1β−1))(eβ−1(ξ)−1+1)u

=uα,β(logβ⁡(1+ξ)|p)ξ(1−v((1+βξ)1β−1))∑ρ=0∞(u+ρ−1ρ)(1−eβ−1(ξ))ρ

=(∑m=0∞Fm,β,p(α)(v)ξmm!)(∑ρ=0∞(u)(ρ)∑l=ρ∞S2,β(l,ρ;−ρ)ξll!)

=(∑m=0∞Fm,β,p(α)(v)ξmm!)(∑l=0∞∑ρ=0l(u)(ρ)S2,β(l,ρ;−ρ)ξll!)

L.H.S=∑m=0∞(∑l=0m∑ρ=0l(ml)Fm−l,β,p(α)(v)(u)(ρ)S2,β(l,ρ;−ρ))ξmm!. (51)

By (51), we obtain the result.

Theorem 3.4. For m ≥ 0 and α∈Z , we have

Fm,β,p(α)(v)=∑l=0m∑a=0m−l∑ρ=0l(ml)(m−la)Dm−l−a,βFa,β(v)p(ρ+1)(1)ρ+1,β(ρ+1)!(ρ+1)αS1,β(l,ρ+1). (52)

Proof. From (45), we observe that

∑m=0∞Fm,β,p(α)(v)ξmm!=uα,β(logβ⁡(1+ξ)|p)ξ(1−v((1+βξ)1β−1))

=1ξ(1−v((1+βξ)1β−1))∑ρ=0∞p(ρ+1)(1)ρ+1,βρ!(ρ+1)αρ!(logβ⁡(1+ξ))ρ+1

=logβ⁡(1+ξ)ξ(1−v((1+βξ)1β−1))∑ρ=0∞p(ρ+1)(1)ρ+1,β(ρ+1)!(ρ+1)α(ρ+1)!(logβ⁡(1+ξ))ρ+1

=logβ⁡(1+ξ)ξ11−v((1+βξ)1β−1)∑ρ=0∞p(ρ+1)(1)ρ+1,β(ρ+1)!(ρ+1)α∑l=ρ∞S1,β(l,ρ+1)ξll!

=(∑s=0∞Ds,βξss!)(∑a=0∞Fa,β(v)ξaa!)(∑l=0∞∑ρ=0mp(ρ+1)(1)ρ+1,β(ρ+1)!(ρ+1)αS1,β(l,ρ+1)ξll!)

=(∑b=0∞∑a=0b(ba)Db−a,βFa,β(v)ξbb!)(∑l=0∞∑ρ=0mp(ρ+1)(1)ρ+1,β(ρ!+1)(ρ+1)αS1,β(l,ρ+1)ξll!)

=∑m=0∞(∑l=0m∑a=0m−l∑ρ=0l(ml)(m−la)Dm−l−a,βFa,β(v)p(ρ+1)(1)ρ+1,β(ρ+1)!(ρ+1)αS1,β(l,ρ+1))ξmm!. (53)

Comparing the coefficients of ξmm! of the above equation, the result is established.

Theorem 3.5. For m ≥ 0 and α∈Z , we have

Fm,β,p(α)(u;v)=mFm−1,β,p(α)(u;v). (54)

Proof. Applying the difference operator ▵β on both sides of Eq. (45), gives

Δβ(∑m=0∞Fm,β,p(α)(u;v)ξmm!)=Δβ(uα,β(logβ⁡(1+ξ)|p)ξ(1−v((1+βξ)1β−1)(1+βξ)uβ)

and then we have

∑m=0∞ΔβFm,β,p(α)(v)ξmm!=uα,β(logβ⁡(1+ξ)|p)ξ(1−v((1+βξ)1β−1)Δβeβu(ξ)

=uα,β(logβ⁡(1+ξ)|p)ξ(1−v((1+βξ)1β−1)eβu(ξ)ξ

=∑m=0∞Fm,β,p(α)(u;v)ξm+1m!. (55)

Therefore, by (55), we obtain the result.

Theorem 3.6. Let m ≥ 0 and α∈Z . Then we have

∂∂uFm,β,p(α)(u;v)=∑ρ=0m(mρ)Fm−ρ,β,p(α)(u;v)(1)ρ,β.

Proof. Applying the operator δδu in both sides of (45), we have

∂∂u(∑m=0∞Fm,β,p(α)(u;v)ξmm!)=∂∂u(uα,β(logβ⁡(1+ξ)|p)ξ(1−v((1+βξ)1β−1)(1+βξ)uβ)

∑m=0∞∂∂uFm,β,p(α)(u;v)ξmm!

=uα,β(logβ⁡(1+ξ)|p)ξ(1−v((1+βξ)1β−1)∂∂u(1+βξ)uβ

=uα,β(logβ⁡(1+ξ)|p)ξ(1−v((1+βξ)1β−1)(1+βξ)uβ(1+βξ)1β

=(∑m=0∞Fm,β,p(α)(u;v)ξmm!)(∑ρ=0∞(1)ρ,βξρρ!)

=∑m=0∞(∑ρ=0m(mρ)Fm−ρ,β,p(α)(u;v)(1)ρ,β)ξmm!. (56)

In view of (56), we complete the proof.

4  Computation of Zeros and Graphical Representation

To understand the structure of the type 2 degenerate poly-Fubini polynomials Fm,β(α)(u;v) , we employ some numerical investigations (using Mathematica ) to examine properties of figures, look and to discover some interesting patterns. First, we give few Fm,β(α)(u;v) (for α = 5, β = 4) as follows:

F0,β(α)(u;v)=1log⁡(4),

F1,β(α)(u;v)=−332log⁡(4)2−12log⁡(4)+ulog⁡(4)+vlog⁡(4),

F2,β(α)(u;v)=781log⁡(4)3+316log⁡(4)2−3u16log⁡(4)2−3v16log⁡(4)2+23log⁡(4)−5ulog⁡(4)+u2log⁡(4)

−4vlog⁡(4)+2uvlog⁡(4)+2v2log⁡(4),

F3,β(α)(u;v)=−2311024log⁡(4)4−718log⁡(4)3+7u27log⁡(4)3+7v27log⁡(4)3−3364log⁡(4)2+27u16log⁡(4)2

−9u232log⁡(4)2+45v32log⁡(4)2−9uv16log⁡(4)2−9v216log⁡(4)2−32log⁡(4)+40ulog⁡(4)−27u22log⁡(4)

+u3log⁡(4)+55v2log⁡(4)−24uvlog⁡(4)+3u2vlog⁡(4)−21v2log⁡(4)+6uv2log⁡(4)+6v3log⁡(4),

F4,β(α)(u;v)=693625log⁡(4)5+231128log⁡(4)4−231u256log⁡(4)4−231v256log⁡(4)4+4927log⁡(4)3−98u27log⁡(4)3

+14u227log⁡(4)3−28v9log⁡(4)3+28uv27log⁡(4)3+28v227log⁡(4)3+158log⁡(4)2−297u16log⁡(4)2

+45u28log⁡(4)2−3u38log⁡(4)2−213v16log⁡(4)2+81uv8log⁡(4)2−9u2v8log⁡(4)2+9v2log⁡(4)2

−9uv24log⁡(4)2−9v34log⁡(4)2+245log⁡(4)−470ulog⁡(4)+204u2log⁡(4)−26u3log⁡(4)+u4log⁡(4)

−291ylog⁡(4)+334uvlog⁡(4)−72u2vlog⁡(4)+4u3vlog⁡(4)+266v2log⁡(4)−132uv2log⁡(4)+12u2v2log⁡(4)

−120v3log⁡(4)+24uv3log⁡(4)+24v4log⁡(4).

Next, we display the shape of Fm,β(α)(u;v) for v = 4, −100 ≤ u ≤ 100 in Fig. 1, for m = 1, 2, …, 8.

Figure 1: Graphs of Fm,β(α)(u;v) (for α = 5, β = 4 (left) and α=7, β=13 (right))

The real and complex zeros of Fm,β(α)(u;v)=0 for v = 4, α = 5, β = 4 are listed in Tab. 1.

The zeros of F8,β(α)(u;v)=0 for u∈C,v=4 are plotted in Fig. 2.

Figure 2: Zeros of F8,β(α)(u;v) (for α = 5, β = 4 (left) and α=7,β=13 (right))

Figure 3: Stacking structure of approximate roots of Fm,β(α)(u;v) (for α = 5, β = 4 (left) and α=7,β=13 (right))

Figure 4: Surface plot of Fm,β(α)(u;v) , m = 1, 3, 5, 8 (for α = 5, β = 4 (left) and α=7,β=13 (right))

From Fig. 2, we note that F8,β(α)(u;v) has Im (u) = 0 reflection symmetry.

The stacking structure of approximate roots of Fm,β(α)(u;v)=0 for v = 4, m = 1, 2, …, 8 is given in Fig. 3.

To provide more clear explanation of the structure of the 2 degenerate poly-Fubini polynomials Fm,β(α)(u;v) , we present the 3D surface for some values of m, these surfaces are displayed in Fig. 4.

5  Conclusions

In this article, we have introduced type 2 degenerate poly-Fubini polynomials and established certain identities of these numbers and polynomials. Further, we defined the degenerate unipoly-Fubini polynomials and derived certain properties of these numbers and polynomials and given multifarious relations including derivative and integral properties. In addition, we have provided relationships between degenerate unipoly-Fubini polynomials and degenerate special polynomials. In the last section, we have these polynomials with the help of Mathematica.

Acknowledgement: 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 work was supported by the Taif University Researchers Supporting Project (TURSP-2020/246), “Taif University, Taif, Saudi Arabia”.

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

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