The main aim of this paper is to introduce and investigate the degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchi polynomials by using monomiality principle and operational methods. We give explicit relations and some identities for the degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchi polynomials

Burak Kurt

Notes on Number Theory and Discrete Mathematics

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Explicit relations on the degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchi polynomials and numbers

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Notes on Number Theory and Discrete MathematicsPrint ISSN 1310–5132, Online ISSN 2367–82752023, Volume 29, Number 3, 486–494DOI: 10.7546/nntdm.2023.29.3.486-494Explicit relations on the degenerate type2-unified Apostol–Bernoulli, Euler and Genocchipolynomials and numbersBurak KurtDepartment of Mathematics and Science Education, Faculty of Education,Akdeniz University, Antalya TR-07058, Turkeye-mail: burakkurt@akdeniz.edu.trReceived: 18 November 2022 Revised: 5 June 2023Accepted: 19 June 2023 Online First: 11 July 2023Abstract: The main aim of this paper is to introduce and investigate the degenerate type 2-unifiedApostol–Bernoulli, Euler and Genocchi polynomials by using monomiality principle andoperational methods. We give explicit relations and some identities for the degenerate type2-unified Apostol–Bernoulli, Euler and Genocchi polynomials.Keywords: Type 2-Bernoulli, Euler and Genocchi polynomials, Degenerate Bernoulli, Euler andGenocchi polynomials, Unified degenerate Apostol–Bernoulli, Euler and Genocchi polynomials,Monomiality principle, Multiplicative operators.2020 Mathematics Subject Classification: 11B68, 11B83, 05A10, 33B11.1 IntroductionThe degenerate versions of special polynomials and numbers initiated by Carlitz [2, 3] haveregained the attention of some mathematics by replacing the usual exponential function in thegenerating function of special polynomials with the degenerate exponential functions. The studyof special polynomials provide many useful identities, their relations and representationsCopyright © 2023 by the Author. This is an Open Access paper distributed under theterms and conditions of the Creative Commons Attribution 4.0 International License(CC BY 4.0). https://creativecommons.org/licenses/by/4.0/associated with special numbers and polynomials. Belingeri et al. [1] and Dattoli et al. [4, 5]considered monomiality principle for the Appell polynomials. Using monomiality principle andoperational methods. Khan et al. [11–13] introduced and investigated the 2-variable Apostol-typepolynomials and degenerate Apostol-type Bernoulli, Euler and Genocchi polynomials.Luo [5], Srivastava [20] and Luo and Srivastava [16] introduced and investigated Apostol–Bernoulli, Euler and Genocchi polynomials and gave some explicit relations and identities forthese polynomials. The type 2-degenerate Apostol–Bernoulli, Euler and Genocchi polynomialshave been introduced in [6–10, 12] and the coauthors of these articles gave recurrence relationsand identities and symmetry properties for these polynomials. Motivated by Khan et al. [11],we define and investigate the type 2-unified degenerate Apostol–Bernoulli, Euler and Genocchipolynomials.We use the usual notations; N, Z, R and C for the set of natural integers, integers, real numbersand complex numbers, respectively. Also, we let N0 := N ∪ {0} and Z− := {−1,−2,−3, · · · }.We begin by introducing the following definition and notation (see also [1–22]). The classicalBernoulli polynomials Bn(x), the classical Euler polynomials En(x) and the classical Genocchipolynomials Gn(x) are defined by the following generating functions, respectively,∞∑n=0Bn(x)tnn!=tet − 1ext, |t| < 2π;∞∑n=0En(x)tnn!=2et + 1ext, |t| < π;∞∑n=0Gn(x)tnn!=2tet + 1ext, |t| < π.When x = 0, we get the Bernoulli numbers Bn(0) := Bn, the Euler numbers En(0) := En andthe Genocchi numbers Gn(0) := Gn, respectively.Kim and Lee [8], Kim D. S. and three coauthors [7] and Kim and two coauthors [6] definedthe type 2-Bernoulli polynomials B∗n (x) and the type 2-Euler polynomials E∗n (x) as∞∑n=0B∗n (x)tnn!=tet − e−text, |t| < π (1)and∞∑n=0E∗n (x)tnn!=2et + e−text, |t| < π2, (2)respectively.Ryoo [19] defined the type 2-Genocchi polynomials as∞∑n=0G∗n (x)tnn!=2tet + e−text, |t| < π2. (3)We define the type 2-Apostol–Bernoulli polynomials B∗(α)n,γ (x) of order α ∈ N, the type2-Apostol-Euler polynomials E∗(α)n,γ (x) of order α ∈ N and the type 2-Apostol–Genocchipolynomials G∗(α)n,γ (x) of order α ∈ N by the following generating functions as follows487∞∑n=0B∗(α)n,γ (x)tnn!=(tγet − e−t)αext, (4)(|t| < π when γ = 1; |t| < |log γ| when γ ̸= 1) ,∞∑n=0E∗(α)n,γ (x)tnn!=(2γet + e−t)αext, (5)(|t| < π2when γ = 1; |t| < |log (−γ)| when γ ̸= 1)and∞∑n=0G∗(α)n,γ (x)tnn!=(2tγet + e−t)αext, (6)(|t| < π2when γ = 1; |t| < |log (−γ)| when γ ̸= 1).Carlitz [2,3] defined the degenerate Bernoulli polynomials Bn (x | λ) and the degenerate Eulerpolynomials En (x | λ) by means of the following generating functions, respectively∞∑n=0Bn (x | λ)tnn!=t(1 + λt)1/λ − 1(1 + λt)x/λ (7)and∞∑n=0En (x | λ)tnn!=2(1 + λt)1/λ + 1(1 + λt)x/λ . (8)The degenerate Genocchi polynomials Gn (x | λ) in [14] are defined as follows∞∑n=0Gn (x | λ)tnn!=2t(1 + λt)1/λ + 1(1 + λt)x/λ , (9)which in the special case when x = 0 in (7), (8) and (9) reduce to the generating function of thedegenerate Bernoulli numbers Bn (λ), the degenerate Euler numbers En (λ) and the degenerateGenocchi numbers Gn (λ), respectively.We define the degenerate type 2-Apostol Bernoulli polynomials B∗(α)n,γ (x | λ) of order α ∈ N,the degenerate type 2-Apostol Euler polynomials E∗(α)n,γ (x | λ) of order α ∈ N and the degeneratetype 2-Apostol Genocchi polynomials G∗(α)n,γ (x | λ) of order α ∈ N by means of the followinggenerating functions as follows∞∑n=0B∗(α)n,γ (x | λ)tnn!=(tγ (1 + λt)1/λ − (1 + λt)−1/λ)α(1 + λt)x/λ , (10)∞∑n=0E∗(α)n,γ (x | λ)tnn!=(2γ (1 + λt)1/λ + (1 + λt)−1/λ)α(1 + λt)x/λ (11)and∞∑n=0G∗(α)n,γ (x | λ)tnn!=(2tγ (1 + λt)1/λ + (1 + λt)−1/λ)α(1 + λt)x/λ , (12)respectively.488We define the generalized degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchipolynomials R∗(α)n,β (x;λ, k, a, b) of order α ∈ N by means of the following generating functions∞∑n=0R∗(α)n,β (x | λ; k, a, b)tnn!=(21−ktkβb (1 + λt)1/λ − ab (1 + λt)−1/λ)α(1 + λt)x/λ(β ∈ C, α, k ∈ N, a, b ∈ R and 0 ̸= λ ∈ R) . (13)For x = 0, we have the degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchi numbersR∗(α)n,β (0 | λ; k, a, b) = R∗(α)n,β (λ; k, a, b).Corollary 1.1. Setting k = a = b = 1 and β = γ in (13), we getR∗(α)n,γ (x | λ; 1, 1, 1) = B∗(α)n,γ (x | λ) .Corollary 1.2. Choosing k = 0, b = −a = 1 and β = γ in (13), we getR∗(α)n,γ (x | λ; 0,−1, 1) = E∗(α)n,γ (x | λ) .Corollary 1.3. Letting k = −2a = b = 1 and 2β = γ in (13), we getR∗(α)n, γ2(x | λ; 1,−12, 1)= G∗(α)n,γ (x | λ) .From the Binomials theorems, we have(1 + λt)x/λ =∞∑n=0(x | λ)ntnn!, (14)where (x | λ)n = x (x− λ) (x− 2λ) · · · (x− (n− 1)λ), (x | λ)0 = 1.The notation of quasi-monomiality was introduced and studied Dattoli [4], Dattoli et al. [5]and Belingeri et al. [1], in details S. Khan et al. in [11–13].According to monomiality principle, a polynomial set {Pn(x)}n∈N is “quasi-monomial”provided there exist two operators M̂ and P̂ playing, respectively the role of multiplicativeand derivative operators for the given set of polynomials. These operator satisfy the followingidentities for all n ∈ N.M̂ {Pn(x)} = Pn+1(x), (15)P̂ {Pn(x)} = nPn−1(x), (16)M̂ P̂ {Pn(x)} = n Pn(x) (17)andPn(x) = M̂(n) {P0(x)} = M̂ {1} . (18)4892 Main theorems and related relationsIn this section, we consider the generalized type 2-unified degenerate Apostol–Bernoulli, Eulerand Genocchi polynomials. By applying the monomiality principle and operator methods, wegive an explicit relations and identities for these polynomials.Theorem 2.1. The following relation holds true:R∗(α)n,β (x | λ; k, a, b) =n∑k=0(nk)R∗(α)n−k,β (λ; k, a, b) (x | λ)k . (19)Proof. From (12) and (13), we find∞∑n=0R∗(α)n,β (x | λ; k, a, b)tnn!=(21−ktkβb (1 + λt)1/λ − ab (1 + λt)−1/λ)α(1 + λt)x/λ=∞∑m=0R∗(α)m,β (λ; k, a, b)tmm!∞∑p=0(x | λ)ptpp!.By using Cauchy product and comparing the coefficient of the both sides, we get (19).Theorem 2.2. The following relation holds true:R∗(α)n,β (x+ y | λ; k, a, b) =n∑k=0(nk)R∗(α)n−k,β (x | λ; k, a, b) R∗(α)k,β (y | λ; k, a, b) . (20)The proof of this theorem is similar to (19). We omit it.Theorem 2.3. The generalized degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchipolynomials R∗(α)n,β (x | λ; k, a, b) are quasi-monimial with respect to the following multiplicativeand derivative operatorsM̂ R∗(α)n,β =xeλDx+αλkeλDx − 1−αe−λDx(βbeDx + abe−Dx)βbeDx − abe−Dx(21)andP̂ R∗(α)n,β =eλDx − 1λ. (22)Proof. Differentiating the generating function (13) partially with respect to t∞∑n=0R∗(α)n+1,β (x | λ; k, a, b)tnn!=∞∑n=0{x1 + λt+αkt− αβb (1 + λt)1−λλ + ab (1 + λt)−1−λλβb (1 + λt)1λ − ab (1 + λt)−1λ}× R∗(α)n,β (x | λ; k, a, b)tnn!. (23)By using the following identities in (23)t{(1 + λt)xλ}=eλDx − 1λ{(1 + λt)xλ}, (24)490we get∞∑n=0R∗(α)n+1,β (x | λ; k, a, b)tnn!=∞∑n=0{xeλDx+αλkeλDx − 1−αe−λDx(βbeDx + abe−Dx)βbeDx − abe−Dx}×R∗(α)n,β (x | λ; k, a, b)tnn!. (25)Comparing the coefficients of both sides of equation (25), we have (21).Using generating function (13) after some simplification, we have (22).Theorem 2.4. The generalized degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchipolynomials R∗(α)n,β (x | λ; k, a, b) satisfy the following differential equation{αk +x(eλDx − 1)λeλDx−αe−λDx(βbeDx + abe−Dx)βbeDx − abe−Dx(eλDx − 1λ)− n}R∗(α)n,β (x | λ; k, a, b) = 0.(26)Proof. Using operators (21) and (22) and in view of the monomiality principle M̂ P̂ {Pn (x)} =nPn (x), we get (26).Theorem 2.5. The following relation holds trueR∗(α)n,β (x | λ; k, a, b)(27)=n+1∑l=0(n+ 1l)R∗(α)n+1−l,β (x | λ; k, a, b){βbabB∗l,βbab(x+ 1 | λ)− B∗l,βbab(x− 1 | λ)}.Proof. From (13) and (10), we can write∞∑n=0R∗(α)n,β (x | λ; k, a, b)tnn!=1t(2(1−k)tkβb (1 + λt)1λ − ab (1 + λt)−1λ)α×{βbabt (1 + λt)x+1λβbab(1 + λt)1λ − (1 + λt)−1λ− t (1 + λt)x−1λβbab(1 + λt)1λ − (1 + λt)−1λ}=1t∞∑n=0R∗(α)n,β (λ; k, a, b)tnn!{βbab∞∑n=0B∗n,βbab(x+ 1 | λ) tnn!−∞∑n=0B∗n,βbab(x− 1 | λ) tnn!}.By using Cauchy product and comparing the coefficients of both sides, we have (27).Theorem 2.6. The following relation holds true:R∗(α)n,β (x | λ; k, a, b) =12n∑l=0(nl)R∗(α)n−l,β (λ; k, a, b)×{βbabE∗l,βbab(x+ 1 | λ) + E∗l,βbab(x− 1 | λ)}. (28)491Proof. From (13) and (11), we find∞∑n=0R∗(α)n,β (x | λ; k, a, b)tnn!=12(21−k tkβb (1 + λt)1λ − ab (1 + λt)−1λ)α(βbab2 (1 + λt)x+1λβbab(1 + λt)1λ + (1 + λt)−1λ+2 (1 + λt)x−1λβbab(1 + λt)1λ + (1 + λt)−1λ)=12∞∑n=0R∗(α)n,β (λ; k, a, b)tnn!{βbab∞∑n=0E∗n,βbab(x+ 1 | λ) tnn!+∞∑n=0E∗n,βbab(x− 1 | λ) tnn!}.Using Cauchy product and comparing the coefficients of tnn!in both sides, we get (28).Theorem 2.7. The following relation holds trueR∗(α)n,β (x | λ; k, a, b) =12n+1∑l=0(n+ 1l)R∗(α)n+1−l,β (λ; k, a, b)×{βbabG∗l,βbab(x+ 1 | λ) + G∗l,βbab(x− 1 | λ)}. (29)3 ConclusionWe introduce and investigate the generalized degenerate type 2-unified Apostol–Bernoulli, Eulerand Genocchi polynomials and give explicit relations by using the monomiality principle. Also,by using to monomiality principle and operators methods to the generalized the degenerate type2-unified Apostol–Bernoulli, Euler and Genocchi polynomials, we have shown give some relationsand identities for these polynomials.References[1] Belingeri, C., Dattoli, G., & Ricci P. E. (2007). The monomialty approach to multi-indexpolynomials in several variables. Georgian Mathematical Journal, 14(1), 53–64.[2] Carlitz, L. (1956). A degenerate Staudt–Clausen theorem. Archiv der Mathematik (Basel),7, 28–33.[3] Carlitz, L. (1979). Degenerate Stirling, Bernoulli and Eulerian numbers. UtilitasMathematica, 15, 51–88.[4] Dattoli, G. (2001). Hermite–Bessel and Laguerre–Bessel functions: A byproduct of themonomiality principle. Advanced Special Functions and Applications, 1, 147–164.492[5] Dattoli, G., Migliorati, M., & Srivastava H. M. (2004). A class of Bessel summationformulas and associated operational methods. 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Explicit relations on the degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchi polynomials and numbers

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