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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
1Department of Mathematics, University of Tabuk, Tabuk, Saudi Arabia
2Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al Khobar, Saudi Arabia
3Department of Mathematics, Hajjah University, Hajjah, Yemen
4Department 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
logβeβ(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)sinπsπ)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.