# Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus

Authors: Nabiullah Khan; Mohd Ghayasuddin; Dojin Kim; Junesang Choi
Journal: Journal of Mathematics
 (2022)

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## Abstract

Among a remarkably large number of various extensions of polynomials and numbers, and diverse introductions of new polynomials and numbers, in this paper, we choose to introduce two new generalizations of some extended Bernoulli polynomials and numbers by using the Mittag–Leffler function and the confluent hypergeometric function. Then, we investigate certain properties and formulas of these newly introduced polynomials and numbers such as explicit representations, addition formulas, integral formulas, differential formulas, inequalities, and inequalities involving their integrals. Also, by using the theory of umbral calculus, five additional formulas regarding these new polynomials are provided. Furthermore, we propose to introduce four generalizations of the extended Euler and Genocchi polynomials. Finally, three natural problems are poised.


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## Body

## 1. Introduction and Preliminaries

We begin by recalling the classical Bernoulli polynomials and numbersBnuu∈ℂ,n∈ℕ0 and Bnn∈ℕ0 (see, e.g., [1], Sections 1.7 and 1.8]):(1)teutet−1=∑n=0∞Bnutnn!,t<2π,and(2)tet−1=∑n=0∞Bntnn!,t<2π.Here and in the following, letℕ, ℤ0−, and ℂ denote the sets of positive integers, nonpositive integers, and complex numbers, respectively. Also, let ℕ0≔ℕ∪0. The generalized Bernoulli polynomials Bnαu and Bnα are defined by means of the generating functions (see, e.g., [1], pp. 83 and 84])(3)tet−1αeut=∑n=0∞Bnαutnn!,t<2π,1α≔1,tet−1α=∑n=0∞Bnαtnn!,t<2π,1α≔1,for arbitrary (real or complex) parameter α. A remarkably large number of polynomials, their extensions, and variants have been presented and investigated due mainly to their usefulness and applications in diverse ways in a wide range of research subjects in science and engineering such as numerical analysis, operator theory, special functions, complex analysis, statistics, sorting, and data compression (see, e.g., [2–17] and the references therein). Take a few instances, Bell [2] proposed and analyzed so-called exponential polynomials, which are polynomials generated by differentiating exponential functions expfx or by expanding the exponential into a power series in x. These polynomials (also called Bell polynomials) are related to Stirling and Bell numbers and arise in many applications. Khan and Usman [8] presented and studied a class of generating functions for Hermite–Bernoulli–Legendre polynomials, the specific examples of which may be connected to particular partial differential equations emerging from physical problems. Simsek [15] developed new families of special numbers with their generating functions, which are related to Golombek’s problem and the calculation of the negative order Euler numbers and applied to certain probability and statistics problems. Bernoulli polynomials and numbers, and their various extensions are obviously among the most important and useful ones (see, e.g., [8, 18–47] and the references therein). In study of Lerch zeta function, Apostol [19] introduced and investigated the following polynomials βnu,λ defined by(4)zeuzλez−1=∑n=0∞βnu,λznn!,λ∈ℂ∖0,z<logλ.Here and elsewhere,log is taken its principal branch. Obviously, βnu,1 is the Bernoulli polynomial Bnu. Luo and Srivastava [26] provided the following Apostol–Bernoulli polynomials βnαu,λ of order α:(5)zλez−1αeuz=∑n=0∞βnαu,λznn!,λ∈ℂ∖0,z<logλ,1α≔1.Natalini and Bernardini [27] introduced and explored the following generalizations of the Bernoulli polynomials:(6)tpet−∑j=0p−1tj/j!eut=∑n=0∞Bnp−1utnn!,u∈ℂ,t<δ,p∈ℕ.Here,δ is the smallest positive real number among the absolute values of complex zeros of(7)et−∑j=0p−1tjj!=0.Clearly,Bn1−1u=Bn0u=Bnu.In this paper, we aim to introduce two new generalizations of some extended Bernoulli polynomials and numbers by using the Mittag–Leffler function (9) and the confluent hypergeometric function (12). Then, we investigate certain properties and formulas of these newly introduced polynomials and numbers such as explicit representations, addition formulas, integral formulas, differential formulas, inequalities, and inequalities involving their integrals. Also, by using the theory of umbral calculus, five additional formulas regarding these new polynomials are presented. Furthermore, we propose to provide four generalizations of the extended Euler and Genocchi polynomials. Finally, three natural problems are poised for the interested researcher.Among the above-cited various generalizations of Bernoulli polynomials and numbers, we recall Kurt’s extension (see [24])(8)tpet−∑j=0p−1tj/j!αeut=∑n=0∞Bnα,p−1utnn!,u∈ℂ,t<δ,p∈ℕ,1α≔1.for the arbitrary (real or complex) parameter α. Here, δ is the same as in (7). Obviously, Bnα,0u=Bnαu. It is remarked in passing that the restrictions in (6) and (8) are suitably modified versions of the original ones.Since Mittag–Leffler (see [28, 29]) introduced the so-called Mittag–Leffler function, Eλz defined by(9)Eλz=∑k=0∞zkΓλk+1,z∈ℂ,ℜλ>0,where Γ is the familiar gamma function given by(10)Γz=∫0∞xz−1e−xdx,ℜz>0,the function and a large number of its extensions (generalizations) have been introduced and applied in a variety of research subjects (see, e.g., [30–34] and the references therein). Clearly,(11)E1z=ez.The confluent hypergeometric functionF11a;b;z is defined (see, e.g., [35], Chaptepr 9], [36], p. 106], [1, 37], pp. 69 and 70], [38], Section 1.3])(12)F11a;b;z=∑n=0∞anbnznn!,a∈ℂ,b∈ℂ∖ℤ0−,z<∞,where the Pochhammer symbol λν is defined (for λ,ν∈ℂ), in terms of the Gamma function Γ, by(13)λν≔Γλ+νΓλ,λ∈ℂ∖ℤ0−=1,ν=0;λ∈ℂ∖0,λλ+1⋯λ+n−1,ν=n∈ℕ;λ∈ℂ,it being understood conventionally that 00≔1 (see, e.g., [1], p. 2 and p. 5]). Obviously,(14)F11a;a;z=ez.


## 2. Two Generalizations of Bernoulli Polynomials and Numbers

In this section, we present two new extensions of Bernoulli polynomials and numbers by using the Mittag–Leffler function (9) and the confluent hypergeometric function (12).Definition 1.
LetEλt be the Mittag–Leffler function (9). Then, extended Bernoulli polynomials associated with the Mittag–Leffler function ℬn,λα,pun∈ℕ0 are defined by means of the following generating function:(15)tp/Γλp+1Eλt−∑j=0p−1tj/Γλj+1αeut=∑n=0∞ℬn,λα,putnn!,p∈ℕ,u∈ℂ,ℜλ>0,t<η,1α≔1,α being an arbitrary (real or complex) parameter, where η is the smallest positive real number among all of the complex zeros of(16)Eλt−∑j=0p−1tjΓλj+1=0.
Here,ℬn,λα,p:=ℬn,λα,p0n∈ℕ0 are called extended Bernoulli numbers associated with the Mittag–Leffler function.Definition 2.
LetF11 be the confluent hypergeometric function (12). Then, extended Bernoulli polynomials associated with the confluent hypergeometric function ℬn,a,bα,pun∈ℕ0 are defined by means of the generating function(17)ap/bpp!tpF11a;b;t−∑j=0p−1aj/bjtj/j!αeut=∑n=0∞ℬn,a,bα,putnn!,p∈ℕ,a,u∈ℂ,b∈ℂ∖ℤ0−,t<ξ,1α≔1,α being an arbitrary (real or complex) parameter, where ξ is the smallest positive real number among all of the complex zeros of(18)F11a;b;t−∑j=0p−1ajbjtjj!=0.
Here,ℬn,a,bα,p≔ℬn,a,bα,p0n∈ℕ0 are called extended Bernoulli numbers associated with the confluent hypergeometric function.It is easy to find from (11) and (14) that(19)ℬn,1α,pu=ℬn,a,aα,pu=p!−αBnα,p−1u.For later use, let us denote(20)Gα;p,λ;u;tandℋα;p,a,b;u;t,the generating functions of the left members in (15) and (17), respectively.


## 3. Formulas and Properties of the Extended Polynomials and Numbers

We present some formulas and properties of the extended Bernoulli polynomials in (15) and (17) as in the following theorems.Theorem 1.
Let the restrictions in Definition1 be satisfied. Also, let n∈ℕ0. Then,(21)ℬn,λα,p=∑n!m1!m2!⋯mn!−1kαk×∏j=1nΓλp+1Γλp+j+1mj,where the sum is over all nonnegative integers m1, m2,…,mn that satisfy m1+2m2+⋯+nmn=n and k=m1+m2+⋯+mn.
We also obtain(22)ℬn,λα,p>0n∈ℕ0,p∈ℕ,α<1−n,λ>0.Proof.
From Definition1, we find that, under the given restrictions,(23)Gα;p,λ;0;t=1+Eλ,pt−α=∑n=0∞ℬn,λα,ptnn!,where(24)Eλ,pt≔∑k=1∞Γλp+1Γλp+k+1tk.
Taylor–Maclaurin series expansion gives(25)ℬn,λα,p=ddtn1+Eλ,pt−α|t=0,n∈ℕ0.
Employing Faà Di Bruno’s formula (see, e.g., [39], p. 5]) in (25), we can obtain the desired result (21).
It may be easy to show from (21) that the inequality (22) holds.The first few ofℬn,λα,p are(26)ℬ0,λα,p=1,ℬ1,λα,p=−αΓλp+1Γλp+1+1,ℬ2,λα,p=−2αΓλp+1Γλp+2+1+αα+1Γλp+1Γλp+1+12,ℬ3,λα,p=−α3Γλp+1Γλp+1+13+6αα+1Γλp+12Γλp+1+1Γλp+2+1−6αΓλp+1Γλp+3+1.Theorem 2.
Let the restrictions in Definition2 be satisfied. Also, let n∈ℕ0. Then,(27)ℬn,a,bα,p=∑n!m1!m2!⋯mn!−1kαk×∏j=1na+pj1+pjb+pjmj,where the sum is over all nonnegative integers m1, m2,…,mn that satisfy m1+2m2+⋯+nmn=n and k=m1+m2+⋯+mn.
We also get(28)ℬn,a,bα,p>0,n∈ℕ0,p∈ℕ,α<1−n,a,b>0.Proof.
From (17), we have(29)ℋα;p,a,b;0;t=1+∑k=1∞a+pk1+pkb+pktk−α=∑n=0∞ℬn,a,bα,ptnn!.Then, a similar process of the proof of Theorem 1 can prove (27) as well as (28). The details are omitted.The first few ofℬn,a,bα,p are(30)ℬ0,λα,p=1,ℬ1,a,bα,p=−αa+pp+1b+p,ℬ2,a,bα,p=−2αa+pa+p+1p+1p+2b+pb+p+1+αα+1a+pp+1b+p2,ℬ3,a,bα,p=−α3a+pp+1b+p3+6αα+1a+p+1a+p2p+2b+p+1p+12b+p2−6αa+p3p+13b+p3.Theorem 3.
Let the restrictions in Definition1 be satisfied. Then,(31)ℬn,λα,pu+v=∑m=0nnmvn−mℬm,λα,pu,n∈ℕ0,u,v∈ℂ.Proof.
From (15), we find(32)∑n=0∞ℬn,λα,pu+vtnn!=evt⋅∑m=0∞ℬm,λα,putmm!=∑n=0∞vntnn!⋅∑m=0∞ℬm,λα,putmm!=∑n=0∞∑m=0nvn−mn−m!m!ℬm,λα,putn,from the first and last sides of which, upon identifying the coefficients of tn, we get the desired formula (31).Theorem 4.
Let the restrictions in Definition2 be satisfied. Then,(33)ℬn,a,bα,pu+v=∑m=0nnmvn−mℬm,a,bα,pu,n∈ℕ0,u,v∈ℂ.Proof.
Here, using (17), the proof would flow in line with that of Theorem 3. The details are omitted.Exchanging the roles ofu and v in (31) and (33), and setting v=0 in the resulting identities, we obtain the following two corollaries.Corollary 1.
Let the restrictions in Definition1 be satisfied. Then,(34)ℬn,λα,pu=∑m=0nnmumℬn−m,λα,p,n∈ℕ0,u∈ℂ.
We find from (34) that ℬn,λα,pu is a polynomial in u of degree nn∈ℕ0.
Together with the results of Theorem1, the first few of ℬn,λα,pu are(35)ℬ0,λα,pu=1,ℬ1,λα,pu=−αΓλp+1Γλp+1+1+u,ℬ2,λα,pu=−2αΓλp+1Γλp+2+1+αα+1Γλp+1Γλp+1+12−2αΓλp+1Γλp+1+1u+u2.Corollary 2.
Let the restrictions in Definition2 be satisfied. Then,(36)ℬn,a,bα,pu=∑m=0nnmumℬn−m,a,bα,p,n∈ℕ0,u∈ℂ.
We find from (36) that ℬn,a,bα,pu is a polynomial in u of degree nn∈ℕ0.
Together with the results of Theorem2, the first few of ℬn,a,bα,pu are(37)ℬ0,λα,pu=1,ℬ1,a,bα,pu=−αa+pp+1b+p+u,ℬ2,a,bα,pu=−2αa+pa+p+1p+1p+2b+pb+p+1+αα+1a+pp+1b+p2−2αa+pp+1b+pu+u2.Theorem 5.
Let the restrictions in Definitions1 and 2 be satisfied. Also let x,y∈ℂ and a,b∈ℂ∖ℤ0−. Then,(38)∫xyℬn,λα,pudu=1n+1ℬn+1,λα,py−ℬn+1,λα,px,n∈ℕ0,(39)∫xyℬn,a,bα,pudu=1n+1ℬn+1,a,bα,py−ℬn+1,a,bα,px,n∈ℕ0.Proof.
Integrating both sides of (15) with respect to u from x to y and simplifying the resulting identity, we get(40)∑n=0∞∫xyℬn,λα,pudutn+1n!=∑n=0∞ℬn,λα,py−ℬn,λα,pxtnn!.
Or, equivalently, in view ofℬ0,λα,pu=1 for any u∈ℂ,(41)∑n=1∞∫xyℬn−1,λα,pudutnn−1!=∑n=1∞ℬn,λα,py−ℬn,λα,pxtnn!,both sides of which, upon equating the coefficients of tn, is easily found to yield formula (38).
Similarly, starting (17), we can get formula (39).Theorem 6.
Let the restrictions in Definition1 be satisfied. Then,(42)∂∂ukℬn,λα,pu=n!n−k!ℬn−k,λα,pu,n,k∈ℕ0,n≥k.Proof.
Differentiating both sides of (15) with respect to u, k times, we obtain(43)tkGα;p,λ;u;t=∑n=0∞∂∂ukℬn,λα,putnn!,in the right-hand side of which term-by-term differentiation is employed, which is guaranteed under the restrictions. We also have(44)tkGα;p,λ;u;t=∑n=0∞ℬn,λα,putn+kn!=∑n=k∞ℬn−k,λα,putnn−k!.
Then, equating the coefficients oftn in the right members of (43) and (44) gives the desired formula (42).Theorem 7.
Let the restrictions in Definition2 be satisfied. Then,(45)∂∂ukℬn,a,bα,pu=n!n−k!ℬn−k,a,bα,pu,n,k∈ℕ0,n≥k.Proof.
Here, using (17), the proof would run parallel to that of Theorem 6. The details are omitted.Theorem 8.
Let the restrictions in Definition1 be satisfied. Then,(46)ℬn,λα+β,pu=∑m=0nnmℬn−m,λβ,pℬm,λα,pu,n∈ℕ0,u∈ℂ,for arbitrary (real or complex) parameters α and β.Proof.
From (15), we obtain(47)∑n=0∞ℬn,λα+β,putnn!=Gα+β;p,λ;u;t=∑n=0∞ℬn,λβ,ptnn!⋅∑m=0∞ℬm,λα,putmm!=∑n=0∞∑m=0nℬn−m,λβ,pn−m!m!ℬm,λα,putn,from the first and last sides of which, upon equating the coefficients of tn, we get the desired formula (46).Theorem 9.
Let the restrictions in Definition2 be satisfied. Then,(48)ℬn,a,bα+β,pu=∑m=0nnmℬn−m,a,bβ,pℬm,a,bα,pu,n∈ℕ0,u∈ℂ,for arbitrary ( real or complex) parameters α and β.Proof.
Here, using (18), the proof would stream in step with that of Theorem 8. The details are omitted.Theorem 10.
Let the restrictions in Definition1 be satisfied. Then,(49)ℬn,λα+β,pu+v=∑m=0nnmℬn−m,λβ,pvℬm,λα,pu,n∈ℕ0,u,v∈ℂ,for arbitrary ( real or complex) parameters α and β.Proof.
Using (15), similarly as in the proof of Theorem 8, we can prove (49). The details are omitted.Theorem 11.
Let the restrictions in Definition2 be satisfied. Then,(50)ℬn,a,bα+β,pu+v=∑m=0nnmℬn−m,a,bβ,pvℬm,a,bα,pu,n∈ℕ0,u,v∈ℂ,for arbitrary (real or complex) parameters α and β.Proof.
Using (17), similarly as in the proof of Theorem 8, we can obtain (50). The details are omitted.We present certain inequalities involving the extended Bernoulli polynomials introduced here as in the following theorem.Theorem 12.
The following inequalities hold:(51)ℬn,λα,pu>0,n∈ℕ0,p∈ℕ,α<1−n,u,λ>0,(52)ℬn,a,bα,pu>0,n∈ℕ0,p∈ℕ,α<1−n,a,b,u>0,(53)∂∂ukℬn,λα,pu>0,n,k∈ℕ0,n≥k,p∈ℕ,α<1−n,u,λ>0,(54)∂∂ukℬn,a,bα,pu>0,n,k∈ℕ0,n≥k,p∈ℕ,α<1−n,a,b,u>0.Proof.
The inequalities here can be easily found by considering (22), (28), (34), (36), (42), and (45).Theorem 13.
The following statements and inequalities hold:(i)
The polynomialsℬn,λα,pu are convex functions on any closed bounded interval x,y with 0≤x<y<∞, provided that(55)n,k∈ℕ0,n≥k≥2,p∈ℕ,α<1−n,u,λ>0.(ii)
The polynomialsℬn,a,bα,pu are convex functions on any closed bounded interval x,y with 0≤x<y<∞, provided that(56)n,k∈ℕ0,n≥k≥2,p∈ℕ,α<1−n,a,b,u>0.(iii)
Hermite–Hadamard-type inequalities are as follows:(57)ℬn,λα,px+y2≤1y−x∫xyℬn,λα,pudu≤ℬn,λα,px+ℬn,λα,py2,0≤x<y<∞,n,k∈ℕ0,n≥k≥2,p∈ℕ,α<1−n,λ>0,(58)ℬn,a,bα,px+y2≤1y−x∫xyℬn,a,bα,pudu≤ℬn,a,bα,px+ℬn,a,bα,py2,0≤x<y<∞,n,k∈ℕ0,n≥k≥2,p∈ℕ,α<1−n,a,b>0.(iv)
Inequalities for the arithmetic and geometric mean are as follows:(59)1x∫0xlogℬn,λα,pudu≤log1x∫0xℬn,λα,pudu,n,k∈ℕ0,n≥k,p∈ℕ,α<1−n,x,λ>0.(60)1x∫0xlogℬn,a,bα,pudu≤log1x∫0xℬn,a,bα,pudu,n,k∈ℕ0,n≥k,p∈ℕ,α<1−n,x,a,b>0.(61)∫01ℬn,λα,puηdu1/η≤∫01ℬn,λα,puμdu1/μ,0<η<μ,n,k∈ℕ0,n≥k,p∈ℕ,α<1−n,λ>0.(62)∫01ℬn,a,bα,puηdu1/η≤∫01ℬn,a,bα,puμdu1/μ,0<η<μ,n,k∈ℕ0,n≥k,p∈ℕ,α<1−n,a,b>0.Proof.
Under the given restrictions, (53) and (54) when k=2 give(63)∂∂u2ℬn,λα,pu>0and∂∂u2ℬn,a,bα,pu>0,which imply the statements (i) and (ii) (see, e.g., [40], pp. 133-135]).
The chains of the inequalities (57) and (58) follow from (i) and (ii) (see, e.g., [41], Eq. (1)]).
The inequalities (59) and (60) can follow from (51) and (52), respectively, by using the inequality for the arithmetic and geometric mean (see, e.g., [42], p. 2]).
For proof of the inequalities (61) and (62), one may consult [(for example) [43], p. 182]. Indeed, it is sufficient to prove the following results: suppose that f:0,1⟶a,b is integrable on 0,1, and fxp is integrable for all 0<p<∞. Then, the following inequalities hold true:(64)∫01fxrdx1/r≤∫01fxdx,0<r≤1,and(65)∫01fxpdx1/p≤∫01xqdx1/q,0<p<q.
To prove (64), recall Jensen’s inequality (see, e.g., [43], p. 179]). let ϕ be convex on a closed interval a,b and g:0,1⟶a,b. If g and ϕ°f are integrable on 0,1, then(66)ϕ∫01gxdx≤∫01ϕ°gxdx.
Chooseϕx=x1/r0≤x<∞,0<r≤1. Since ϕ″x=1/r1/r−1x1/r−2≥0 for all x∈0,∞, ϕ is convex on 0,∞. Then, choose gx=fxr. Now, applying Jensen’s inequality (66) to these chosen ϕx and gx gives the inequality (64).
For (65), one finds from (64) that(67)∫01fxqrdx1/r≤∫01fxqdx.Setting r=p/q∈0,1 in (67) provides(68)∫01fxpdxq/p≤∫01fxqdx.Since x1/qx∈0,∞ is an increasing function, taking 1/q th power on both members of the inequality (68) preserves the inequality to produce the inequality (65).
Now, the inequalities (61) and (62) are obviously particular instances of (65).
It is passing in noting that the methods of proof of this theorem have already been employed in [13].


## 4. Certain Identities from Umbral Calculus

In this section, we try to provide some new identities which are not given in the previous sections by using the theory of umbral calculus (see, e.g., [4, 44–46]). For our purpose, some notations with modified ones and certain known facts are recalled and introduced. In what follows, let P denotes the algebra of formal power series in the variable t over the field C of characteristic zero, say, ℂ the field of complex numbers. If ft and gt in P satisfy(69)ft=∑k=0∞aktk,a0=0,a1≠0andgt=∑k=0∞bktk,b0≠0,then ft and gt are said to be a delta series and an invertible series, respectively. With each pair of a delta series ft and an invertible series gt, there exists a unique sequence snu of polynomials gratifying the orthogonality conditions (see [45], p. 17, Theorem 2.3.1])(70)gtftk|snu=n!δn,k,n,k∈ℕ0,where δn,k is the Kronecker delta function defined by δn,k=1n=k and δn,k=0n≠k, if and only if the sequence snu is generated by means of the following function:(71)1gf¯tevf¯t=∑k=0∞skvk!tk,for all v in ℂ, where f¯t=f−1t is the compositional inverse of ft (see [45], p. 18, Theorem 2.3.4]). Here, the operator ⋅|⋅ remains the same as in [45] [Chapter 2]. That is,(72)gtftk|snu≔gDufDuksnu|u=0,where Du≔∂/∂u. The sequence snu satisfying (70) or (71) is called the Sheffer sequence for gt,ft, or snu is Sheffer for gt,ft. In particular, if snu is Sheffer for 1,ft, then snu is called the associated sequence for ft, or snu is associated to ft; if snu is Sheffer for gt,t, then snu is called the Appell sequence for gt, or snu is Appell for gt (see [45], p. 17).In view of (20), (15) can be rewritten as(73)Gu;t=Gteut=∑n=0∞ℬn,λα,putnn!,where as follows:(74)Gα;p,λ;u;t=Gu;tandGα;p,λ;0;t≔Gt.In light of the result in Theorem1, the Gt is an invertible series. So the sequence ℬn,λα,pun∈ℕ0 is Appell for Gt.There are many equivalent statements for the Appell sequence (see [45], pp. 26–28). In this connection, some of the equivalent statements for the Appell sequence ℬn,λα,pun∈ℕ0 are chosen to be given in the following theorems. Here, we will just show Theorem 15 and will leave proofs of the other theorems to the reader who is interested.Theorem 14.
(The polynomial expansion theorem) Let the restrictions in Definition1 be assumed. Then, for any polynomial pu, we obtain(75)pu=∑k=0∞GDupkuu=0k!ℬk,λα,pu,where pku is the k th derivative of pu.Theorem 15.
(The conjugate representation) Let the restrictions in Definition1 be assumed. Then,(76)ℬn,λα,pu=∑k=0nnkGDu−1un−ku=0uk.
Here and in the following,Gt−1 is the reciprocal of Gt.Proof.
For simplicity, letsnu≔ℬn,λα,pu. Since the sequence snun∈ℕ0 is Appell for Gt, it follows from (71) that(77)1Gtevt=∑k=0∞skvk!tk,for all v in ℂ. Applying the right member of (77) to un, in view of (72), offers(78)∑k=0∞skvk!tk|un=∑k=0∞skvk!Dukun|u=0=snv.
Applying the left member of (77) to un, in terms of (72), gives(79)1Gtevt|un=∑k=0∞1k!vkGt−1tk|un=∑k=0∞1k!GDu−1Dukunu=0vk=∑k=0nnkGDu−1un−ku=0vk.
Note thatDukun=nn−1⋯n−k+1un−k. Since gt is an invertible series, its reciprocal has the Maclaurin series expansion with the nonzero constant term; that is, gt−1=1/gt=∑k=0∞cktkc0≠0. Finally, matching the right members of (78) and (79) provides(80)snv=∑k=0nnkGDu−1un−ku=0vk,for all v in ℂ. Since (80) holds for all v∈ℂ, the identity (76) follows.Theorem 16.
Let the restrictions in Definition1 be assumed. Then,(81)ℬn,λα,pu=GDu−1un.Theorem 17.
Let the restrictions in Definition1 be assumed. Then, for any ht in P,(82)hDuℬn,λα,pu=∑k=0nnkhDuℬk,λα,puu=0un−k.Theorem 18.
(The multiplication theorem) Let the restrictions in Definition1 be assumed. Then, for any constant κ∈ℂ∖0,(83)ℬn,λα,pκu=κnGDuGDu/κℬn,λα,pu,n∈ℕ0.


## 5. Concluding Remarks and Further Observations

Among a remarkably large number of various extensions of polynomials and numbers, and diverse introductions of new polynomials and numbers, in this paper, we introduced two new generalizations of some extended Bernoulli polynomials and numbers by using the Mittag–Leffler function and the confluent hypergeometric function. Then, we investigated certain properties and formulas of these new introduced polynomials and numbers such as explicit representations, addition formulas, integral formulas, differential formulas, inequalities, and inequalities involving their integrals. Also, by using the theory of umbral calculus, five additional formulas regarding these new polynomials were presented.Like the generalization of the extended Bernoulli polynomials in (8), Khan et al. [5] also introduced certain generalizations of the extended Euler polynomials Enα,p−1u and the extended Genocchi polynomials Gnα,p−1u as follows (here, a detailed history of extensions of the polynomials is given):(84)2pet+∑l=0p−1tl/l!αeut=∑n=0∞Enα,p−1utnn!,(85)2ptpet+∑l=0p−1tl/l!αeut=∑n=0∞Gnα,p−1utnn!.Here, as in this paper, we propose to introduce the following further generalizations of the extended Euler polynomials (84) and the extended Genocchi polynomials (85) by using the Mittag–Leffler function (9) and the confluent hypergeometric function (12) as follows:(86)2pEλt+∑j=0p−1tj/Γλj+1αeut=∑n=0∞Εn,λα,putnn!,p∈ℕ,u∈ℂ,ℜλ>0,t<ρ,1α≔1,α being arbitrary (real or complex) parameter, where ρ is the smallest positive real number among all of the complex zeros of(87)Eλt+∑j=0p−1tjΓλj+1=0,(88)2pF11a;b;t+∑j=0p−1aj/bjtj/j!αeut=∑n=0∞En,a,bα,p−1utnn!,p∈ℕ,a,u∈ℂ,b∈ℂ∖ℤ0−,t<σ,1α:=1,α being an arbitrary (real or complex) parameter, where σ is the smallest positive real number among all of the complex zeros of(89)F11a;b;t+∑j=0p−1ajbjtjj!=0,(90)2ptpEλt+∑j=0p−1tj/Γλj+1αeut=∑n=0∞Gn,λα,p−1utnn!,p∈ℕ,u∈ℂ,ℜλ>0,t<ρ,1α≔1,α being an arbitrary (real or complex) parameter.(91)2ptpF11a;b;t+∑j=0p−1aj/bjtj/j!αeut=∑n=0∞Gn,a,bα,p−1utnn!,p∈ℕ,a,u∈ℂ,b∈ℂ∖ℤ0−,t<σ,1α≔1,α being an arbitrary (real or complex) parameter.We conclude this paper by posing the following natural questions which are left to the interested researchers:(i)
Deduce certain identities for the polynomialsℬn,a,bα,pu, n∈ℕ0 in (17), as those in Section 4(ii)
Give certain properties and formulas for the four new polynomials (86), (88), (90), and (91) as those in Sections 3 and 4(iii)
Find or approximate the complex zeros in equations (7), (16), (18), (87), and (89)



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