A note on type 2 degenerate poly-Frobenius-Genocchi polynomials Waseem A. Khan
Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Kingdom of Saudi Arabia
E-mail: [email protected]
Abstract. In [18], Kim et al. introduced the degenerate poly-Bernoulli polyno-mials by using polyexponential function. In this paper, we study the degenerate Frobenius-Genocchi polynomials, which are called the type 2 degenerate poly-Frobenius-Genocchi polynomials, by means of polyexponential function. Then, we de-rive some useful relations and properties. We dede-rive type 2 degenerate poly-Frobenius-Genocchi polynomials equal a linear combination of the degenerate Frobenius-poly-Frobenius-Genocchi polynomials and Stirling numbers of the first kind. Furthermore, we introduce type 2 degenerate unipoly-Frobenius-Genocchi polynomials by means of unipoly function and derive explicit multifarious properties.
Keywords: polylogarithm function, Frobenius-Genocchi polynomials, degenerate poly-Frobenius-Euler polynomials.
2010 Mathematics Subject Classification: 05A19, 11B68, 11B83, 11B95. 1. Introduction
Recently, Kim and his research team (see [7-18]) have studied the degenerate versions of special numbers and polynomials actively. This idea provides a powerful tool to define special numbers and polynomials of their degenerate versions. We can say that the notion of degenerate version from a special class of polynomials because of their great applicability. The most important application of these polynomials is in the theory of finite differences, analytic number theory, applications in classical analysis, and statistics. Despite the applicability of special functions in classical anal-ysis and statistics, they also arise in communications systems, quantum mechanics, nonlinear wave propagation, electric circuit theory. electromagnetic theory, etc.
As is well known, the classical Bernoulli, Euler and Genocchi polynomials are respectively, defined by (see [4, 5, 6])
t et−1e
xt= ∞ ∑
j=0
Bj(x)
tj
j!,|t|<2π, 2
et+ 1e xt=
∞ ∑
j=0
Ej(x)
tj
j!,|t|< π (1.1)
and
2t et+ 1e
xt= ∞ ∑
j=0
Gj(x)
tj
j!,|t|< π.
In the case when x= 0,Bj =Bj(0), Ej =Ej(0) andGj =Gj(0) are respectively, called the Bernoulli, Euler and Genocchi numbers.
Foru∈Cwithu̸= 1, the Frobenius-Genocchi polynomialsGF
n(x;u) are defined by
(1−u)t et−u e
xt= ∞ ∑
n=0
GFn(x;u)t n
n!,(see [18). (1.2)
In the case whenx = 0, GFn(u) =GFn(0;u) are called the Frobenius-Genocchi numbers.
Obviously
GFn(x;−1) =Gn(x).
For k ∈ Z, Kim-Kim [7] defined the modified polyexponential function, as an inverse to the polylogarithm function by
Eik(x) = ∞ ∑
n=1
xn
nk(n−1)!. (1.3)
It is worthy to note thate(x,1|k) =x1Eik(x) and Ei1(x) =ex−1.
For any non-zeroλ∈ R(or C), the degenerate exponential function is defined by
exλ(t) = (1 +λt)xλ, eλ(t) = (1 +λt) 1
λ. (1.4)
By binomial expansion, we get
exλ(t) = ∞ ∑
n=0 (x)n,λ
tn
n!,(see [12, 13]), (1.5) where (x)0,λ= 1, (x)n,λ= (x−λ)(x−2λ)· · ·(x−(n−1)λ),(n≥1).
Note that
lim λ−→0e
x λ(t) =
∞ ∑
n=0
xnt
n
n! =e xt.
In [1, 2], Carlitz introduced the degenerate Bernoulli polynomials given by
t eλ(t)−1
exλ(t) =
t
(1 +λt)λ1 −1
(1 +λt)xλ =
∞ ∑
n=0
βn(x;λ,)
tn
n!. (1.6) The casex= 0,βn(λ) =βn(0;λ) are called the degenerate Bernoulli numbers.
Note that
lim
λ−→0βn(x;λ) =Bn(x).
In [17], Kim et al. considered the the degenerate Genocchi polynomials given by
2z eλ(z) + 1
euλ(z) = ∞ ∑
j=0
Gj,λ(u)
zj
j!. (1.7)
In the case whenu= 0,Gj,λ=Gj,λ(0) are called the degenerate Genocchi numbers.
Very recently, Kim et al. [18] introduced the degenerate poly-Bernoulli polyno-mials defined by
Eik(log(1 +t))
eλ(t)−1
exλ(t) = ∞ ∑
n=0
βn,λ(k)(x)t n
n!. (1.8)
Here,β(n,λk) =βn,λ(k)(0) are called the degenerate poly-Bernoulli numbers.
1−u
(1 +λt)1λ −u
(1 +λt)λx =
∞ ∑
n=0
hn,λ(x|u)
tn
n!, (1.9)
so that
hn,λ(x|u) = n ∑
m=0 (
n m
)
hm,λ(u) (x
λ
)
n−m
, (n≥0).
When x = 0, hn,λ(u) = hn,λ(0|u) are called the degenerate Frobenius-Euler numbers, (see [8, 7]).
Note that
lim
λ−→0hn,λ(x|u) =Hn(x|u),(n≥0).
It is well known that the Stirling numbers of the first kind are defined by
(x)n= n ∑
l=0
S1(n, l)xl,(see [6, 7]), (1.10)
where (x)0 = 1, and (x)n =x(x−1)· · ·(x−n+ 1),(n≥1). From (1.10), it is easily to see that
1
k!(log(1 +t)) k =
∞ ∑
n=k
S1(n, k)
tn
n!, (k≥0),(see [6, 9, 13-15]). (1.11) In the inverse expression to (1.11), the Stirling numbers of the second kind are defined by
xn= n ∑
l=0
S2(n, l)(x)l,(see [1,6, 7]). (1.12)
From (1.12), it is easily to see that 1
k!(e
t−1)k = ∞ ∑
n=l
S2(n, l)
tn
n!,(see [8, 11, 13, 14]). (1.13) The generating function of the degenerate Stirling numbers of the second kind [13-19] are defined by
1
n!(eλ(t)−1) n=
∞ ∑
l=n
S2,λ(l, n)
tl
l!,(n≥0). (1.14) In this article, we consider the generating function for the type 2 degenerate Frobenius-Genocchi polynomials, which are called the type 2 degenerate poly-Frobenius-Genocchi polynomials, by means of polyexponential function and derive some properties of those polynomials. In the last section, we define type 2 degenerate unipoly-Frobenius-Genocchi polynomials by means of unipoly function and includes several properties.
2. Type 2 degenerate poly-Frobenius-Genocchi polynomials
Letλ, u∈Cwith u̸= 1 and k∈Z, by using the polyexponential function, we consider the type 2 degenerate poly-Frobenius-Genocchi polynomials are defined by means of the following generating function
Eik(log(1 + (1−u)t))
eλ(t)−u
exλ(t) = ∞ ∑
n=0
G(n,λF,k)(x;u)t n
In case whenx= 0 in (2.1), G(n,λF,k)(u) =G(n,λF,k)(0;u) are called the type 2 de-generate poly-Frobenius-Genocchi numbers.
Fork= 1 in (2.1), we get
(1−u)t eλ(t)−u
exλ(t) = ∞ ∑
n=0
GFn,λ(x;u)t n
n!, (2.2)
whereGF
n,λ(x;u) are called the degenerate Frobenius-Genocchi polynomials. Obviously
lim λ−→0
(
Eik(log(1 + (1−u)t))
eλ(t)−u
)
exλ(t) = ∞ ∑
n=0 lim λ−→0G
(F,k) n,λ (x;u)
tn
n!
= (1−u)t
et−u e xt=
∞ ∑
n=0
G(nF)(x;u)t n
n!. (2.3)
Thus, by (2.1) and (2.3), we have lim λ−→0G
(F,k)
n,λ (x;u) =G (F,k)
n (x;u),(n≥0) (2.4) whereG(nF,k)(x;u) are called the type 2 poly-Frobenius-Genocchi polynomials.
It is clear that
G(n,λF,1)(x;u) =G(n,λF)(x;u), G(n,λF,1)(x;−1) =Gn(x)(see [4]), lim
λ−→0G (F,1)
n,λ (x;u) =G (F)
n (x;u),(n≥0),(see [18]). Theorem 2.1. Forn≥0, we have
n ∑
l=0 (
n l
)∑∞
m=0 1 (m+ 1)k−1
∞ ∑
m=l
S1(l+ 1, m+ 1)(x)n−l,λ
(1−u)l+1
l+ 1
= n ∑
m=0 (
n m
)
G(nF,k−m,λ) (x|u)(1)m,λ−uG (F,k)
n,λ (x|u). (2.5)
Proof. From (2.1), we have
Eik(log(1 + (1−u)t))exλ(t) =eλ(t) ∞ ∑
n=0
G(n,λF,k)(x|u)t n
n!−u ∞ ∑
n=0
G(n,λF,k)(x|u)t n
n!
= ∞ ∑
m=0 (1)m,λ
tm m!
∞ ∑
n=0
G(n,λF,k)(x|u)t n
n! −u ∞ ∑
n=0
G(n,λF,k)(x|u)t n
n!
= ∞ ∑
n=0 ( n
∑
m=0 (
n m
)
G(nF,k−m,λ) (x|u)(1)m,λ−uG (F,k) n,λ (x|u)
)
tn
n!. (2.6) On the other hand,
Eik(log(1 + (1−u)t))exλ(t)
= ( ∞
∑
n=0 (x)n,λ
tn
n! ) ( ∞
∑
m=1
(log(1 + (1−u)t))m (m−1)!mk
= (∞
∑
n=0 (x)n,λ
tn n! ) ( ∞ ∑ m=0 1 (m+ 1)k−1
∞ ∑
l=m+1
S1(l, m+ 1)
(1−u)ltl
l! ) = (∞ ∑ n=0 (x)n,λ
tn n! ) (∞ ∑ l=0 ∞ ∑ m=0 1 (m+ 1)k−1
∞ ∑
m=l
S1(l+ 1, m+ 1)(1−u) l+1
l+ 1
tl l! ) = ∞ ∑ n=0 ( n ∑ l=0 ( n l )∑∞ m=0 1 (m+ 1)k−1
∞ ∑
m=l
S1(l+ 1, m+ 1)(x)n−l,λ
(1−u)l+1
l+ 1 )
tn
n!. (2.7) Comparing the coefficients of tnn! on both sides of equation (2.6) and (2.7), we obtain the following theorem.
Theorem 2.2. Forn≥0, we have
G(n,λF,k)(x;u) = n ∑ m=0 ( n m )
G(nF,k−m,λ) (u)(x)m,λ. (2.8)
Proof. From (2.1), we have ∞
∑
n=0
G(n,λF,k)(x;u)t n
n! = (
Eik(log(1 + (1−u)t))
eλ(t)−u
)
exλ(t)
= ∞ ∑
n=0
G(n,λF,k)(u)t n
n! ∞ ∑
m=0 (x)m,λ
tm m! L.H.S= ∞ ∑ n=0 n ∑ m=0 ( n m )
G(nF,k−m,λ) (u)(x)m,λ
tn
n!. (2.9)
Therefore, by (2.1) and (2.9), we require at the desired result. Theorem 2.3. Fork∈Zandn≥0, we have
G(n,λF,k)(x;u) = n ∑ l=0 ( n l )∑∞ m=0 1 (m+ 1)k
∞ ∑
m=l
S1(l+ 1, m+ 1)(1−u)l
l+ 1 G
F
n−l,λ(x;u). (2.10) Proof. By using equations (1.11), (2.2) and (2.1), we have
∞ ∑
n=0
G(n,λF,k)(x;u)t n
n! = (
Eik(log(1 + (1−u)t))
eλ(t)−u
)
exλ(t)
= e x λ(t)
eλ(t)−u ∞ ∑
m=1
(log(1 + (1−u)t))m (m−1)!mk
= e x λ(t)
eλ(t)−u ∞ ∑
m=0
(log(1 + (1−u)t))m+1
m!(m+ 1)k
= e x λ(t)
eλ(t)−u ∞ ∑
m=0 1 (m+ 1)k
∞ ∑
n=m+1
S1(n, m+ 1)((1−u)t) n
n!
= (1−u)t
eλ(t)−u
exλ(t) ∞ ∑
m=0 1 (m+ 1)k
∞ ∑
m=n
S1(n+ 1, m+ 1)(1−u)n
n+ 1
tn n! = ∞ ∑ n=0
GFn,λ(x;u)
tn n! ∞ ∑ m=0 1 (m+ 1)k
∞ ∑
m=l
S1(l+ 1, m+ 1)(1−u)l
l+ 1
L.H.S = ∞ ∑
n=0 ( n
∑
l=0 (
n l
)∑∞
m=0 1 (m+ 1)k
∞ ∑
m=l
S1(l+ 1, m+ 1)(1−u)l
l+ 1 G
F
n−l,λ(x;u) )
tn n!. (2.11) By comparing the coefficients of tnn!, we arrive at the desired result (2.10). Corollary 2.1. Fork∈Zandn≥0, we have
G(n,λF,k)(u) = n ∑
l=0 (
n l
)∑∞
m=0 1 (m+ 1)k
∞ ∑
m=l
S1(l+ 1, m+ 1)(1−u)l
l+ 1 G
F n−l,λ(u). Corollary 2.2. Forn≥0, we have
GFn,λ(x;u) = n ∑
l=0 (
n l
)∑∞
m=0 ∞ ∑
m=l
S1(l+ 1, m+ 1)(1−u)l
l+ 1 G
F
n−l,λ(x;u). Corollary 2.3. Forn≥0, we have
Gn,λ(x) = n ∑
l=0 (
n l
)∑∞
m=0 ∞ ∑
m=l
S1(l+ 1, m+ 1)2l
l+ 1 Gn−l,λ(x). In particular,
n ∑
l=0 (
n l
)∑∞
m=0 ∞ ∑
m=l
S1(l+ 1, m+ 1)2l
l+ 1 Gn−l,λ(x) = 0. It is well-known from ([3]) that
(
t
log(1 +t) )r
(1 +t)x−1= ∞ ∑
n=0
Bn(n−r+1)(x)t n
n!,(r∈C), (2.12) whereBn(r)(x) are called the higher-order Bernoulli polynomials which are given by the generating function
(
t et−1
)r
ext= ∞ ∑
n=0
Bn(r)(x)t n
n!. Theorem 2.4. Forn≥0, we have
G(n,λF,2)(u) = l ∑
l=0 (
n l
)
(1−u)lBl l
l+ 1 G F
n−l,λ(u). (2.13)
Proof. Using (1.3), we first consider the following expression
d
dxEik(log(1 + (1−u)x)) = d dx
∞ ∑
n=1
(log(1 + (1−u)x))n (n+ 1)!nk
= 1−u
(1 + (1−u)x) log(1 + (1−u)x) ∞ ∑
n=1
(log(1 + (1−u)x))n (n+ 1)!nk−1
= 1−u
(1 + (1−u)x) log(1 + (1−u)x)Eik−1(log(1+(1−u)x). (2.14) From (2.14),k≥1, we have
∞ ∑
n=0
G(n,λF,k)(u)x n
n! =
(1−u)k−1
eλ(x)−u ∫ x
0
1
×
∫ t
0
1
(1 + (1−u)t) log(1 + (1−u)t)· · · ∫ t
0
| {z }
k−2−times
t
(1 + (1−u)t) log(1 + (1−u)t)dtdt...dt.
Hence, we require
∞ ∑
n=0
G(n,λF,2)(u)x n
n! =
(1−u)
eλ(x)−u ∫ x
0
(1−u)t
(1 + (1−u)t) log(1 + (1−u)t). (2.15)
= (1−u)
eλ(x)−u ∫ x
0 ∞ ∑
n=0
(1−u)nBnnt
n
n!dt
= (1−u)x
eλ(x)−u ∞ ∑
n=0
(1−u)nBn n
n+ 1
xn
n!
= (∞
∑
n=0
GFn,λ(u)x n
n! ) ( ∞
∑
n=0
(1−u)nBnn n+ 1
xn n!
)
L.H.S= ∞ ∑
n=0 ( l
∑
l=0 (
n l
)
(1−u)lBll l+ 1 G
F n−l,λ(u)
)
xn
n!. (2.16) By (15) and (16), we require at the desired result. Thus, we complete the
proof.
Theorem 2.5. Letk≥1 andm∈N∪{0},s∈C, we have
χk,u,ν(−m) = (1−u)−m−1(−1)mG(m,νF,k)(u). (2.17) Proof. Letk≥1, be an integer. Fors∈C, we define the functionχk,ν(s) as
χk,ν(s) = 1 Γ(s)
∫ ∞
0
zs−1
eν(z)−u
Eik(log(1 + (1−u)z))dz. (2.18)
In view of calculation above thatχk,ν(s) is holomorphic function for ℜ(s)>0 because of the comparison test as Eik(log(1 + (1−u)z)) ≤ E1k(log(1 + (1−u)z)) with the assumption (1−u)t≥0. From (2.18), we note that
χk,ν(s) =
(1−u)s−1 Γ(s)
∫ ∞
0
zs−1 eν(z)−u
Eik(log(1 + (1−u)z))dz
= (1−u) s−1 Γ(s)
∫ 1
0
zs−1
eν(z)−u
Eik(log(1 + (1−u)z))dz
+(1−u) s−1 Γ(s)
∫ ∞
1
zs−1 eν(z)−u
Eik(log(1 + (1−u)z))dz. (2.19) The second integral converges absolutely for any s∈Cand hence, the second term on the right hand side vanishes at non-positive integers. That is,
lim s→−m
(1−Γ(us))s−1∫1∞eν(zz)−uEik(log(1 + (1−u)z))dz
≤(1−u)−m−1
Γ(−m) M = 0, (2.20) since
Γ(s)Γ(1−s) = π sin(πs)
On the other hand, forℜ(s)>0, the first integral in (2.20) can be written as (1−u)s−1
Γ(s) ∫ 1
0
zs−1
eν(z)−u
= (1−u) s−1 Γ(s)
∞ ∑
n=0
G(n,λF,k)(u)
n! ∫ 1
0
zn+s−1dz
= (1−u) s−1 Γ(s)
∞ ∑
n=0
G(n,λF,k)(u)
n! 1
n+s. (2.21)
which defines an entire function of s. Thus, we may include that χk,ν(s) can be continued to an entire function ofs. Further, from (2.20) and (2.21), we obtain
χk,ν,u(−m) = lim s→−m
(1−u)s−1 Γ(s)
∫ 1
0
zs−1
eν(z)−u
Eik(log(1 + (1−u)z))dz
= lim s→−m
(1−u)s−1 Γ(s)
∞ ∑
r=0
G(r,νF,k)
s+r
1
r!
=· · ·+ 0 +· · ·+ 0 + lim s→−m
(1−u)s−1 Γ(s)
1
s+m
G(m,νF,k)(u)
m! + 0 + 0 +· · · (2.22)
= lim s→−m
(1−u)s−1 Γ(s)
Γ(1−s) sinπs π
G(m,νF,k)(u)
m! = (1−u)
−m−1Γ(1 +m) cos(πm)G (F,k) m,ν
m!
= (1−u)−m−1(−1)mG(m,νF,k)(u).
Thus, we complete the proof of this theorem. Theorem 2.6. Forn≥0, we have
G(n,λF,k)(x+y;u) = n ∑
m=0 (
n m
)
G(nF,k−m,λ) (x;u)(y)m,λ. (2.23)
Proof. From (2.1), we have ∞
∑
n=0
G(n,λF,k)(x+y;u)t n
n! = (
Eik(log(1 + (1−u)t))
eλ(t)−u
)
exλ+y(t)
= ( ∞
∑
n=0
G(n,λF,k)(x;u)t n
n! ) ( ∞
∑
m=0 (y)m,λ
tm
m! )
= ∞ ∑
n=0 ( n
∑
m=0 (
n m
)
G(nF,k−m,λ) (x;u)(y)m,λ )
tn
n!. (2.24)
By comparing the coefficients ofTn, we obtain the result (2.23). Theorem 2.7. Forn≥0, we have
G(n,λF,k)(x+ 1;u) = n ∑
m=0 (
n m
)
G(nF,k−m,λ) (x;u)(1)m,λ. (2.25)
Proof. By (2.1), we observe that
∞ ∑
n=0 [
G(n,λF,k)(x+ 1;u)−G(n,λF,k)(x;u) ]tn
n! = (
Eik(log(1 + (1−u)t))
eλ(t)−u
)
exλ(t) [eλ(t)−1]
= ∞ ∑
n=0 n ∑
m=0 (
n m
)
G(nF,k−m,λ) (x;u)(1)m,λ
tn n!−
∞ ∑
n=0
G(n,λF,k)(x;u)t n
Theorem 2.8. Forn≥0, we have
G(n,λF,k)(x;u) = n ∑ m=0 m ∑ q=0 ( n m ) (x)qS
(2)
λ (m, q)G (F,k)
n−m,λ(u). (2.27)
Proof. From (2.1), we have ∞
∑
n=0
G(n,λF,k)(x;u)t n
n! = (
Eik(log(1 + (1−u)t))
eλ(t)−u
)
exλ(t)
= (
Eik(log(1 + (1−u)t))
eλ(t)−u
)
[eλ(t)−1 + 1] x
= (
Eik(log(1 + (1−u)t))
eλ(t)−u
) ∑∞
q=0 (x)q
∞ ∑
l=q
Sλ(2)(l, q)t l l! = ∞ ∑ n=0 ( n ∑ m=0 m ∑ q=0 ( n m ) (x)qS
(2)
λ (m, q)G (F,k) n−m,λ(u)
)
tn
n!. (2.28)
By (2.1) and (2.19), we obtain (2.27).
Theorem 2.9. Forn≥0, we have
G(n,λF,k)(x+α|u) = n ∑ l=0 l ∑ m=0 ( n l )
xmm!S2,λ(l+α, m+α)F (F,k)
n−l,λ(u). (2.29)
Proof. Replacingxbyx+αin (2.1), we have ∞
∑
n=0
G(n,λF,k)(x+α;u)t n
n! = (
Eik(log(1 + (1−u)t))
eλ(t)−u
)
exλ+α(t)
= (
Eik(log(1 + (1−u)t))
eλ(t)−u
)
eαλ(t) ( ∞
∑
m=0
xm(eλ(t)−1)m )
= (
Eik(log(1 + (1−u)t))
eλ(t)−u
) (
eαλ(t) ∞ ∑
m=0
xmm!(eλ(t)−1) m
m! )
= (
Eik(log(1 + (1−u)t))
eλ(t)−u
) (
eαλ(t) ∞ ∑
m=0
xmm! ∞ ∑
l=m
S2,λ(l, m)
tl l! ) = ∞ ∑ n=0
G(n,λF,k)(u)t n n! (∞ ∑ l=0 l ∑ m=0
xmm!S2,λ(l+α, m+α)
tl l! ) L.H.S = ∞ ∑ n=0 ( n ∑ l=0 l ∑ m=0 ( n l )
xmm!S2,λ(l+α, m+α)G (F,k) n−l,λ(u)
)
tn
n!. (2.30) Therefore, by (2.1) and (2.30), we get the result (2.29).
3. Type 2 degenerate unipoly-Frobenius-Genocchi polynomials
Let p be any arithmetic function which is a real or complex valued function defined on the set of positive integers N. Kim-Kim [7] defined the unipoly function attached to polynomialsp(x) by
uk(x|p) = ∞ ∑
n=1
p(n)
nk x
Moreover,
uk(x|1) = ∞ ∑
n=1
xn
nk = Lik(x),(see [3]), (3.2) is the ordinary polylogaritm function.
By using (3.1), we define the type 2 degenerate unipoly-Frobenius-Genocchi polynomials by
uk(log(1 + (1−u)t)|p)
eλ(t)−u
exλ(t) = ∞ ∑
n=0
G(n,λ,pF,k)(x;u)t n
n!. (3.3) In the case whenx= 0, G(n,λ,pF,k)(u) =G(n,λ,pF,k)(0;u) are called the type 2 degenerate unipoly-Frobenius-Genocchi numbers. Let us takep(n) = Γ(1n). Then, we have
∞ ∑
n=0
G(n,λ,F,k)1 Γ
(x;u)t n
n! =
uk(log(1 + (1−u)t)|Γ1p)
eλ(t)−u
exλ(t)
= 1
eλ(t)−u
exλ(t) ∞ ∑
m=1
(log(1 + (1−u)t))m
mk(m+ 1)!
= Eik(log(1 + (1−u)t))
eλ(t)−u
exλ(t)
= ∞ ∑
n=0
G(n,λF,k)(x;u)t n
n!. (3.4)
Thus, we have
G(n,λ,F,k)1 Γ
(x;u) =G(n,λF,k)(x;u).
Theorem 3.1. Letn∈Nandk∈Z. Then we have
G(n,λ,pF,k)(u) = n ∑
l=0 l ∑
m=0 (
n l
)
p(m+ 1)(m+ 1)! (m+ 1)k
S1(l+ 1, m+ 1)
l+ 1 (1−u) lGF
n−l,λ(u). (3.5) In particular,
G(n,λ,F,k)1 Γ
(u) = n ∑
l=0 l ∑
m=0 (
n l
)
m+ 1 (m+ 1)k
S1(l+ 1, m+ 1)
l+ 1 (1−u) lGF
n−l,λ(u). (3.6)
Proof. From (3.3), we get ∞ ∑
n=0
G(n,λ,pF,k)(u)t n
n! =
uk(log(1 + (1−u)t)|p)
eλ(t)−u
= 1
eλ(t)−u ∞ ∑
m=1
p(m)
mk (log(1 + (1−u)t)) m
= 1
eλ(t)−u ∞ ∑
m=0
p(m+ 1)(m+ 1)! (m+ 1)k
∞ ∑
l=m+1
S1(l, m+ 1)(1−u) ltl
l!
= (1−u)t
eλ(t)−u ∞ ∑
m=0
p(m+ 1)(m+ 1)! (m+ 1)k
∞ ∑
m=l
S1(l+ 1, m+ 1)
l+ 1 (1−u) lt
l
= (∞
∑
n=0
GFn,λ(u)t n
n! ) ( ∞
∑
m=0
p(m+ 1)(m+ 1)! (m+ 1)k
∞ ∑
m=l
S1(l+ 1, m+ 1)
l+ 1 (1−u) ltl
l! )
= ∞ ∑
n=0 ( n
∑
l=0 l ∑
m=0 (
n l
)
p(m+ 1)(m+ 1)! (m+ 1)k
S1(l+ 1, m+ 1)
l+ 1 (1−u) lGF
n−l,λ(u) )
tn
n!. (3.7) Therefore, by comparing the coefficients on both sides of (3.7), we obtain the following
theorem.
Theorem 3.2. Letn≥0 andk∈Z. Then we have
G(n,λ,pF,k)(x;u) = n ∑
m=0 m ∑
q=0 (
n m
) (x)qS
(2)
λ (m, q)G (F,k)
n−m,λ,p(u). (3.8)
Proof. Using (3.3), we observe that ∞
∑
n=0
G(n,λ,pF,k)(x;u)t n
n! = (
uk(log(1 + (1−u)t))
eλ(t)−u )
exλ(t)
= (
uk(log(1 + (1−u)t))
eλ(t)−u
)
[eλ(t)−1 + 1]x
= (
uk(log(1 + (1−u)t))
eλ(t)−u
) ∑∞
q=0 (x)q
∞ ∑
l=q
S(2)λ (l, q)t l
l!
L.H.S= ∞ ∑
n=0 ( n
∑
m=0 m ∑
q=0 (
n m
) (x)qS
(2)
λ (m, q)G (F,k) n−m,λ,p(u)
)
tn
n!. (3.9) By comparing the coefficients oftn, we obtain the result (3.8).
Theorem 3.3. Letn≥0 andk∈Z. Then we have
G(n,λ,pF,k)(x;u) = n ∑
m=0 (
n m
)
G(nF,k−m,λ,p) (u)(x)m,λ. (3.10)
Proof. From (3.3), we observe that ∞
∑
n=0
G(n,λ,pF,k)(x;u)t n
n! =
uk(log(1 + (1−u)t))
eλ(t)−u
exλ(t)
= ( ∞
∑
n=0
G(n,λ,pF,k)(u)t n
n! ) ( ∞
∑
m=0 (x)m,λ
tm
m! )
L.H.S = ∞ ∑
n=0 ( n
∑
m=0 (
n m
)
G(nF,k−m,λ,p) (u)(x)m,λ )
tn
n!. (3.11) By comparing coefficients on both sides of (3.11), we obtain the following
4. Conclusions
Motivated by the definition of the type 2 degenerate poly-Bernoulli polynomi-als introduced by Kim et al. [18], in the present paper, we have considered a class of new generating function for the Frobenius-Genocchi polynomials, called the type 2 degenerate poly-Frobenius-Genocchi polynomials, by means of the polyexponen-tial function. Then, we have derived some useful relations and properties. We have showed that the type 2 degenerate poly-Frobenius-Genocchi polynomials equal a lin-ear combination of the degenerate Frobenius-Genocchi polynomials and degenerate Stirlings numbers of the first and second kind. In a special case, we have given a re-lation between the type 2 degenerate Frobenius-Genocchi polynomials and Bernoulli polynomials of ordern. Moreover, inspired by the definition of uniBernoulli poly-nomials introduced by Kim-Kim [7] we have introduced the type 2 degenerate unipoly-Frobenius-Genocchi polynomials by means of unipoly function and given multifarious properties including degenerate Stirling numbers of the second kind and degenerate Frobenius-Genocchi polynomials.
Author Contributions: All authors contributed equally to the manuscript and typed, read, and approved final manuscript.
Conflict of Interest: The authors declare no conflict of interest. Funding: None.
Acknowledgements: None.
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