A New Family Of Degenerate Poly-Genocchi Polynomials With Its Certain .
HindawiJournal of Function SpacesVolume 2021, Article ID 6660517, 8 pageshttps://doi.org/10.1155/2021/6660517Research ArticleA New Family of Degenerate Poly-Genocchi Polynomials with ItsCertain PropertiesWaseem A. Khan ,1 Rifaqat Ali ,2 Khaled Ahmad Hassan Alzobydi,2 and Naeem Ahmed31Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O. Box 1664,Al Khobar 31952, Saudi Arabia2Department of Mathematics, College of Science and Arts, Muhayil, King Khalid University, P.O. Box 9004,61413 Abha, Saudi Arabia3Department of Civil Engineering, College of Engineering, Qassim University, Unaizah, Saudi ArabiaCorrespondence should be addressed to Waseem A. Khan; wkhan1@pmu.edu.saReceived 27 November 2020; Revised 7 February 2021; Accepted 22 May 2021; Published 15 June 2021Academic Editor: Tuncer AcarCopyright 2021 Waseem A. Khan et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.In this paper, we introduce a new type of degenerate Genocchi polynomials and numbers, which are called degenerate polyGenocchi polynomials and numbers, by using the degenerate polylogarithm function, and we derive several properties of thesepolynomials systematically. Then, we also consider the degenerate unipoly-Genocchi polynomials attached to an arithmeticfunction, by using the degenerate polylogarithm function, and investigate some identities of those polynomials. In particular, wegive some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.1. IntroductionIn [1, 2], Carlitz initiated a study of degenerate versions ofsome special polynomials and numbers, namely, the degenerate Bernoulli and Euler polynomials and numbers. Kimet al. [3–5] have studied the degenerate versions of specialnumbers and polynomials actively. These ideas provide apowerful tool in order to define special numbers and polynomials of their degenerate versions. The notion of degenerateversion forms a special class of polynomials because of theirgreat applicability. Despite the applicability of special functions in classical analysis and statistics, they also arise in communication systems, quantum mechanics, nonlinear wavepropagation, electric circuit theory, electromagnetic theory,etc. In particular, Genocchi numbers have been extensivelystudied in many different contexts in such branches of mathematics as, for instance, elementary number theory, complexanalytic number theory, differential topology (differentialstructures on spheres), theory of modular forms (Eisensteinseries), p-adic analytic number theory (p-adic L-functions),and quantum physics (quantum groups). The works ofGenocchi numbers and their combinatorial relations havereceived much attention [6–11]. In the paper, we focus on anew type of degenerate poly-Genocchi polynomial andnumbers.The aim of this paper is to introduce a degenerate versionof the poly-Genocchi polynomials and numbers, the socalled new type of degenerate poly-Genocchi polynomialsand numbers, constructing from the degenerate polylogarithm function. We derive some explicit expressions andidentities for those numbers and polynomials.The classical Euler polynomials En ðxÞ and the classicalGenocchi polynomials Gn ðxÞ are, respectively, defined bythe following generating functions (see [12–22]):2 xt tne, ExðÞnet 1n!n 0jt j π,ð1Þ
2Journal of Function Spaces2t xt tne, GxðÞnet 1n!n 0jt j π:ð2Þ In the case when x 0, En ð0Þ En and Gn ð0Þ Gn are,respectively, called the Euler numbers and Genocchinumbers.The degenerate exponential function [23, 24] is definedbyexλ ðt Þ ð1 λt Þx/λ ,tn log ð1 t Þ:n 1 nLi1 ðt Þ logλ ð1 t Þ λn 1 ð1Þn,1/λn 1Note that xn t n ext :n!n 0lim ð1 λt Þx/λ ð4Þeλ ðlogλ ðt ÞÞ logλ ðeλ ðt ÞÞ t: 22tnexλ ðt Þ ð1 λt Þx/λ En ðx ; λÞ :1/λe λ ðt Þ 1n!ð1 λt Þ 1n 0ð6Þlim logλ ð1 t Þ ð 1Þn 1λ 0n 1 ð λÞn 1 ð1Þn,1/λ nx ðk ℤ, jxj 1Þ:ðn 1Þ!nkn 1ðxÞn,λ xðx λÞ ðx ðn 1ÞλÞ ðn 1Þ,From (11) and (14), we getwith the assumption ðxÞ0,λ 1.In [5], Kim et al. considered the degenerate Genocchipolynomials given byð8ÞIn the case when x 0, Gn,λ Gn,λ ð0Þ are called thedegenerate Genocchi numbers.For k ℤ, the polylogarithm function is defined by apower series in t, which is also a Dirichlet series in k (see[25, 26]): tnt2 t3 t ðjt j 1Þ:k2k 3kn 1 nð9ÞThis definition is valid for arbitrary complex order k andfor all complex arguments t with jtj 1 : it can be extended tojtj 1 by analytic continuation.tn log ð1 t Þ:n!l k ,λ ð x Þ It is clear that (see [27, 28])ð7Þð12Þð13ÞThe degenerate polylogarithm function [3] is defined byKim and Kim to beIn the case when x 0, Bn,λ ð0Þ Bn,λ are called thedegenerate Bernoulli numbers and En,λ ð0Þ En,λ are calledthe degenerate Euler numbers.Let ðxÞn,λ be the degenerate falling factorial sequencegiven byLik ðt Þ ð11ÞIt is noteworthy to mention that ntttex ðt Þ ð1 λt Þx/λ βn ðx ; λÞ ,1/λe λ ðt Þ 1 λn!ð1 λt Þ 1n 0ð5Þ 2ttnexλ ðt Þ Gn,λ ðxÞ :eλ ðt Þ 1n!n 0tn,n!being the inverse of the degenerate version of the exponentialfunction eλ ðtÞ as has been shown below:In [1, 2], Carlitz introduced the degenerate Bernoulli anddegenerate Euler polynomials defined by ð10ÞFor λ ℝ, Kim and Kim [3] defined the degenerate version of the logarithm function, denoted by logλ ð1 tÞ, as follows (see [4]):ð3Þe1λ ðt Þ eλ ðt Þ ðλ ℝÞ:λ 0It is noticed that xn Lik ðxÞ:kn 1 nlim lk,λ ðxÞ λ 0 ð λÞn 1 ð1Þn,1/λ nx logλ ð1 xÞ:n!n 1l1,λ ðxÞ ð14Þð15Þð16ÞVery recently, Kim and Kim [3] introduced the new typeof degenerate version of the Bernoulli polynomials and numbers, by using the degenerate polylogarithm function as follows: lk,λ ð1 eλ ð t ÞÞ xtnð kÞe λ ð t Þ βn ,λ ð x Þ :1 eλ ð t Þn!n 0ðkÞð17ÞðkÞWhen x 0, β j,λ β j,λ ð0Þ are called the new type ofdegenerate poly-Bernoulli numbers.The degenerate Stirling numbers of the first kind [24] aredefined by 1tnðlogλ ð1 t ÞÞk S1,λ ðn, kÞ ðk 0Þ:k!n!n kð18Þ
Journal of Function Spaces3It is clear thatlim S1,λ ðn, kÞ S1 ðn, kÞ,λ 0ð19Þcalling the Stirling numbers of the first kind given by (see[29, 30]) 1tnðlog ð1 t ÞÞk S1 ðn, kÞ ðk 0Þ:k!n!n klogarithm function which is called the degenerate polyGenocchi polynomials as follows.For k ℤ, we define the new type of degenerate Genocchinumbers, which are called the degenerate poly-Genocchinumbers, as n2ðkÞ tlk,λ ð1 eλ ð t ÞÞ Gn,λ :e λ ðt Þ 1n!n 0ð20Þð24ÞNote thatThe degenerate Stirling numbers of the second kind[31] are given by (see [2, 13–22, 25–32]) 1tnðeλ ðt Þ 1Þk S2,λ ð j, kÞ ðk 0Þ:k!n!n kð21Þlim S2,λ ðn, kÞ S2 ðn, kÞ,n 0 tn22ttn l1,λ ð1 eλ ð t ÞÞ G n, λ :eλ ðt Þ 1 n 0n! eλ ðt Þ 1n!ð25Þð22Þstanding for the Stirling numbers of the second kind givenby means of the following generating function (see [1–8,12–38]): k 1 ttne 1 S2 ðn, kÞ ðk 0Þ:k!n!n kð 1ÞThus, we have (see [6])Note here thatλ 0 Gn,λð1ÞGn,λ Gn,λ ðn 0Þ:Now, we consider the new type of degenerate Genocchipolynomials which are called the degenerate poly-Genocchipolynomials defined byð23ÞThis paper is organized as follows. In Section 1, werecall some necessary stuffs that are needed throughoutthis paper. These include the degenerate exponential functions, the degenerate Genocchi polynomials, the degenerate Euler polynomials, and the degenerate Stirlingnumbers of the first and second kinds. In Section 2, weintroduce the new type of degenerate poly-Genocchi polynomials by making use of the degenerate polylogarithmfunction. We express those polynomials in terms of thedegenerate Genocchi polynomials and the degenerate Stirling numbers of the first kind and also of the degenerateEuler polynomials and the Stirling numbers of the firstkind. We represent the generating function of the degenerate poly-Genocchi numbers by iterated integrals fromwhich we obtain an expression of those numbers in termsof the degenerate Bernoulli numbers of the second kind.In Section 3, we introduce the new type of degenerateunipoly-Genocchi polynomials by making use of thedegenerate polylogarithm function. We express those polynomials in terms of the degenerate Genocchi polynomialsand the degenerate Stirling numbers of the first kind andalso of the degenerate Euler polynomials and the Stirlingnumbers of the first kind and second kind.ð26Þ 2lk,λ ð1 eλ ð t ÞÞ xtnð kÞe λ ð t Þ G n ,λ ð x Þ :eλ ðt Þ 1n!n 0ðkÞð27ÞðkÞIn the case when x 0, Gn,λ Gn,λ ð0Þ. Using equation(27), we see ðkÞ G n ,λ ð x Þn 0t n 2lk,λ ð1 eλ ð t ÞÞ x eλ ðt Þeλ ðt Þ 1n! ð kÞ Gm,λm 0 nn 0m 0tm tn ð x Þ n ,λm! n 0n!ð28Þ!ðkÞ ðn/mÞGm,λ ðxÞn m,λtn:n!Therefore, by equation (28), we obtain the followingtheorem.Theorem 1. Let n be a nonnegative integer. Then,ð kÞnðkÞGn,λ ðxÞ ðn/mÞGm,λ ðxÞn m,λ :ð29Þm 02. New Type of Degenerate Genocchi Numbersand PolynomialsFrom (27), we note thatIn this section, we define the new type of degenerate Genocchi numbers and polynomials by using the degenerate poly- Gn,λ ðxÞ n 0ð kÞtn2t1 ex ðt Þ l ð1 eλ ð t ÞÞ,n! eλ ðt Þ 1 λ t k,λð30Þ
4 Journal of Function SpacesðkÞ G n, λ ð x Þn 0From (27), we havem 1tn2t1 ð λÞ ð1Þm,1/λ ex ð t Þ ð1 eλ ð t ÞÞmn! eλ ðt Þ 1 λ t m 1 ðm 1Þ!mk m 12t1 ð λÞ ð1Þm,1/λexλ ðt Þ eλ ð t Þ 1t m 1mk 1 ðk Þ Gn , λn 0lt2t1 ex ðt Þl! eλ ðt Þ 1 λ t! lλm 1 ð1Þm,1/λ ð 1Þl 1tlS2,λ ðl, mÞ k 1l!ml 1 m 1!!!m 1l t n 1 l 1 λ ð1Þm,1/λ ð 1Þt l 1 Gn,λ ðxÞS2,λ ðl 1, mÞ k 1n! t l 0 m 1mðl 1Þ!n 0!! ! l 1 m 1λ ð1Þm,1/λ ð 1Þl S2,λ ðl 1, mÞ t ltn Gn,λ ðxÞn!l!l 1mk 1n 0l 0 m 1!!m 1l nn l 1 λ ð1Þ 1ðÞSl 1,mtnðÞ2 ,λm,1/λ :Gn l,λ ðxÞn!l 1mk 1n 0 l 0l m 1 ð 1Þl m S2,λ ðl, mÞl mð31ÞTherefore, by equations (30) and (31), we get the following theorem.xn22 l ð1 eλ ð xÞÞ eλ ð x Þ 1n! eλ ðxÞ 1 k,λð x 1 λðð t 1 λ neλ ð t Þ t eλ1 λ ð t Þeλ ð t ÞðkÞ xtdtdt dt Gn,λ n!0 1 eλ ð t Þ 0 1 eλ ð t Þ0 1 eλ ð t Þn 0 fflfflfflfflfflfflfflffl}2 eλ ðxÞ 1ðkÞGn,λ ðxÞ nnl 0l ð t 1 λð t 1 λe1 λeλ ð t Þeλ ð t Þλ ð t Þ tdtdt dt :0 1 eλ ð t Þ 0 1 eλ ð t Þ0 1 eλ ð t Þ �fflfflfflfflfflfflfflfflfflfflfflfflffl}ðk 2Þ timesð36ÞFor k 2 in (36) and using [3] (Eq. (27)), we get ð2Þ Gn,λn 0xn2 n! eλ ðxÞ 12 e λ ðx Þ 1Theorem 2. Let n be a nonnegative integer. Then,! λm 1 ð1Þm,1/λ ð 1Þl S2,λ ðl 1, mÞ Gn l,λ ðxÞ:l 1mk 1m 1l 1 n 0n 0 ðkÞ Gn,λ ðxÞn 0tn2 ex ðt Þl ð1 eλ ð t ÞÞ,n! eλ ðt Þ 1 λ k,λð33Þðx β j,λ ð1 λÞð 1Þ j0 j 0tjdtj!ð37Þxn β j,λ ð1 λÞxj ð 1Þ jn! j 0 j 1j!!β j,λ ð1 λÞ xn: ðn/jÞð 1Þ Gn j,λn!j 1n 0 j 0Using equations (27) and (6), we seeðkÞte1 λλ ð t Þdt0 1 eλ ð t Þβ j,λ ð1 λÞ2xxj ð 1Þ jeλ ðxÞ 1 j 0 j 1j! ðx Gn,λð32Þ Gn,λ ðxÞðk 2Þ timesðxnjTherefore, by equation (37), we get the followingtheorem.Theorem 4. Let n be a nonnegative integer. Then, ð λÞm 1 ð1Þm,1/λtn2 ex ðt Þ ð1 eλ ð t ÞÞmn! eλ ðt Þ 1 λ m 1 ðm 1Þ!mknð kÞðkÞGn,λ ðxÞ ðn/mÞGm,λ ðxÞn m,λ : ð λÞm 1 ð1Þm,1/λ2exλ ðt Þ eλ ðt Þ 1mk 1m 1ð38Þm 0 tl2 e x ðt Þl! eλ ðt Þ 1 λl m! lλm 1 ð1Þm,1/λ ð 1Þl 1tl Sl,mðÞ2,λk 1l!ml 1 m 1!! !m 1 lnλ ð1Þm,1/λ ð 1Þl 1ttl En,λ ðxÞS2,λ ðl, mÞk 1n!l!mn 0l 1 m 1!!m 1l 1 nlnλ ð1Þm,1/λ ð 1Þtn :S2,λ ðl, mÞEn l,λ ðxÞ k 1n!mn 1 l 1l m 1 ð 1Þl m S2,λ ðl, mÞð34ÞBy equations (33) and (34), we obtain the followingtheorem.Theorem 3. Let n be a nonnegative integer. Then,ðkÞG n ,λ ð x Þ nnl 1l !λm 1 ð1Þm,1/λ ð 1Þl 1S2,λ ðl, mÞEn l,λ ðxÞ:mk 1m 1l ð35ÞIn general, by equation (37), we see ðkÞ Gn,λn 0ð x 1 λðxn2eλ ð t Þ t e1 λλ ð t Þ n! eλ ðxÞ 1 0 1 eλ ð t Þ 0 1 eλ ð t Þð t 1 λeλ ð t Þ tdtdt dt0 1 eλ ð t Þ βn1 ,λ ð1 λÞ βn2 ,λ ð1 λÞ1 n1 1 n1 n2 1n ,n , ,n n n1 !n2 ! nk 1 !12k 1βnk 1 ,λ ð1 λÞ2x ð xÞn1 ,n2 , ,nk 1n1 nk 1 1e λ ðx Þ 1! nβn1 ,λ ð1 λÞn ð 1Þn1 1n1 ,n2 , ,nk n n1 , n2 , , nkn 0βnk 1 ,λ ð1 λÞβn ,λ ð1 λÞxn 2 Gn,λ :n!n1 n 2 1n1 nk 1 1ð39ÞBy equation (39), we obtain the following theorem.
Journal of Function Spaces5Theorem 7. For n ℕ, we haveTheorem 5. Let k ℤ and n 0, we haveð kÞG n ,λ ð 1Þn !n n1 ,n2 , ,nk nn1 , n2 , , nkβnk 1 ,λ ð1 λÞβn1 ,λ ð1 λÞn1 1βn 2 ,λ ð 1 λ Þ G :n1 n2 1n1 nk 1 1 n,λnð40Þλm 1tm2lk,l ð1 eλ ð t ÞÞ ð1 eλ ðt ÞÞ m!m 0 tjðkÞðkÞ G j,λ G j,λ ð1Þ:j!j 1ð1Þλ 0ð41Þλ 0 ð λÞr 1 ð1Þr,1/λð1 eλ ð t ÞÞrkr 1ðÞ!rr 13. Degenerate Unipoly-Genocchi Numbersand Polynomials ð λÞm 1 ð1Þm,1/λ 1ð1 eλ ð t ÞÞmk 1m!mm 1 2 jð λÞr 1 ð1Þr,1/λ j r t Sj,r 1ðÞðÞ2 ,λj!rk 1r 1j r!j ð 1Þ j 1 ð1Þr,1/λ r 1tj 2 :λSj,rðÞ2,λj!rk 1j 1 r 1 pðnÞ nx ðk ℤÞ:kn 1 nuk ðx pÞ 2 ð42ÞTherefore, by equations (41) and (42), we get the following theorem.Theorem 6. Let k ℤ and j 1. Then,ð43ÞFrom equations (27) and (14), we see ð λÞm 1 ð1Þm,1/λð1 eλ ð t ÞÞmkm 1!mðÞm 1 ð λÞm 1 ð1Þm,1/λð1 eλ ð t ÞÞmm!m 1 2 2 ð λÞm 1 ð1Þm,1/λ S2,λ ðn, mÞð 1Þn mn mnn 1m 1 2 ð 1Þn 1!ð1Þm,1/λ λm 1S2,λ ðn, mÞ xn Lik ðxÞkn 1 nuk ð x 1 Þ ð48Þis the ordinary polylogarithm function.In [8], Lee and Kim defined the degenerate unipoly function attached to polynomials pðxÞ as follows:uk,λ ðx pÞ pðiÞð λÞi 1 ð1Þi,1/λi 1ikxi :ð49ÞIt is worthy to note that2t 2l1,l ð1 eλ ð t ÞÞ 2 ð47ÞMoreover (see [25]), jm 1ð46ÞLet p be any arithmetic function which is a real or complexvalued function defined on the set of positive integers ℕ.Kim and Kim [29] defined the unipoly function attached topolynomials pðxÞ by2lk,l ð1 eλ ð t ÞÞ 2 ið1Þr,1/λ r 11 h ðkÞðkÞG j,λ G j,λ ð1Þ ð 1Þ j 1 k 1λ S2,λ ð j, rÞ:2r 1 rð 1Þlim Gn,λ Gn , lim Gn,λ ðxÞ Gn ðxÞ:ðkÞGm,λOn the other hand, ð45Þwhere δn,k is Kronecker’s symbol.Note thatFrom (27), we observe that ð 1Þn 1 ð1Þm,1 λm 1 S2,λ ðn, mÞ δn,1 , 1 lk,λ ðxÞu k ,λ x Γð50Þis the degenerate polylogarithm function.Now, we define the degenerate unipoly-Genocchi polynomials attached to polynomials pðxÞ byntn!tn:n!ð44ÞBy comparing the coefficients on both sides of (44), weobtain the following theorem. 2uk,λ ð1 eλ ð t Þ pÞ xtnð kÞeλ ðt Þ Gn,λ,p ðxÞ :e λ ðt Þ 1n!n 0ðkÞðkÞð51ÞIn the case when x 0, Gn,λ,p Gn,λ,p ð0Þ are called thedegenerate unipoly-Genocchi numbers attached to p.
6Journal of Function SpacesBy, equations (51) and (56), we obtain the followingtheorem.From (51), we see ðkÞ Gn,λ,1/Γn 0 tn21 uk,λ 1 eλ ð t Þ Γn! eλ ðt Þ 1 ð λÞ2 eλ ðt Þ 1 r 1r 1ð1Þr,1/λ ð1 eλ ð t ÞÞrk ðr 1Þ!Theorem 9. Let n be a nonnegative integer. Then,rð52ÞðkÞ n2ðkÞ t lk,λ ð1 eλ ð t ÞÞ Gn,λ :eλ ðt Þ 1n!n 0nnl 1lGn,λ,p ðxÞ !pðmÞλm 1 ð1Þm,1/λ ð 1Þl 1 m!S2,λ ðl, mÞEn l,λ ðxÞ:mkm 1l ð57ÞThus, by (52), we haveð kÞð kÞGn,λ,1 Gn,λ :Γ From (51), we have ðkÞ Gn,λ,p ðxÞn 0From (6), (49), and (51), we getð53ÞðkÞ Gn,λ,p ðxÞn 0tn2exλ ðt Þ u ð1 eλ ð t Þ pÞn! eλ ðt Þ 1 k,λtn2exλ ðt Þ2t eλ ðt Þ 1 x 1 u ð1 eλ ð t Þ pÞ e ðt Þeλ ðt Þ 1 eλ ðt Þ 1 λ tn! eλ ðt Þ 1 k,λ pðmÞð λÞm 1 ð1Þm,1/λð1 eλ ð t ÞÞmmkm 1 m 12exλ ðt Þ 1 pðmÞð λÞ ð1Þm,1/λ ð1 eλ ð t ÞÞmeλ ðt Þ 1 t m 1mk 2t1 pðmÞð λÞ ð1Þm,1/λ m!ex ð t Þ eλ ðt Þ 1 λ t m 1mkm 1 tl2t1 exλ ðt Þ ð 1Þl m S2,λ ðl, mÞ et l 1l!t 1λð Þl m!!m 1l 1l λ ð1Þm,1/λ ð 1Þ m!tltn 1 Sl,mGxðÞðÞ2,λn,λl!n! tmkm 1n 0!! l 1l 1pðmÞλm 1 ð1Þm,1/λ ð 1Þl m!tS2,λ ðl 1, mÞ kmðl 1Þ!l 0 m 1! !! l 1pðmÞλm 1 ð1Þm,1/λ ð 1Þl m! S2,λ ðl 1, mÞ t ltn Gn,λ ðxÞkl 1n!l!mn 0l 0 m 1!!l nn l 1 pðmÞλm 1 ð1Þ 1m!ðÞSl 1,mtnðÞ2,λm,1/λGn l,λ ðxÞ :l 1n!mkn 0 l 0l m 1nl 0l! l 12tex/2λ/2 ð2t Þðe ðt Þ 1Þ eλ/2 ð2t Þ 1 λl 0 m 1! !pðmÞλm 1 ð1Þm,1/λ ð 1Þl m! S2,λ ðl 1, mÞ t l l!l 1mkl 0 m 1!! x 2n t nið1Þt i 1,λ βn,λ/22 n!n 0i 0 i 1 i!! ! l 1pðmÞλm 1 ð1Þm,1/λ ð 1Þl m! S2,λ ðl 1, mÞ t l l!l 1mkl 0 m 1!! i x 2n t ni ð1Þi l 1,λ βn,λ/2 2n!i l 1n 0i 0 l 0 l! !l 1pðmÞλm 1 ð1Þm,1/λ ð 1Þl m! S2,λ ðl 1, mÞ t i i!l 1mkm 1! ! n ii l 1n n 0 i 0 l 0l m 1i Theorem 8. Let n be a nonnegative integer. Then,n pðmÞð 1Þl ðλÞm 1 ð1Þm,1/λ m!S2,λ ðl 1, mÞmk!! x 2 n t ntlti ð1Þi,λ βn,λ/2 2 n!i!ðl 1Þ!n 0i 1Therefore, by equation (54), we get the followingtheorem.ðkÞm 12texλ ðt Þ1 pðmÞð λÞ ð1Þm,1/λðeλ ðt Þ 1Þ ð1 eλ ð t ÞÞm2t m 1mkeλ ðt Þ 1 ð54ÞGn,λ,p ðxÞ pðmÞλm 1 ð1Þm,1/λ ð 1Þl m! S2,λ ðl 1, mÞGn l,λ ðxÞ:l 1mkm 1l 1 ð1Þi l 1,λ pðmÞλm 1 ð1Þm,1/λ ð 1Þl m! S2,λ ðl 1, mÞl 1ði l 1Þmk x t nn iÞ : 2 βn i,λ/22 n!ð55Þ Using equations (49) and (51), we see ðkÞ G n, λ ð x Þn 0l 1ð58Þ pðmÞð λÞm 1 ð1Þm,1/λtn2 ex ðt Þ ð1 eλ ð t ÞÞmn! eλ ðt Þ 1 λ m 1mk pðmÞð λÞ ð1Þm,1/λ m!2ex ðt Þ eλ ðt Þ 1 λ m 1mkm 1 ð 1Þl ml mTherefore, by (58), we obtain the following theorem.tl2S2,λ ðl, mÞ ex ð t Þl! eλ ðt Þ 1 λ!pðmÞλm 1 ð1Þm,1/λ ð 1Þl 1 m!tlSl,m ðÞ2,λkl!ml 1 m 1!! !m 1l 1 lnpðmÞλ ð1Þm,1/λ ð 1Þ m!ttl E n, λ ð x ÞSl,mðÞ2,λkn!l!mn 0l 1 m 1!!m 1l 1 nlnjpðmÞλ ð1Þm,1/λ ð 1Þ m!t S2,λ ðl, mÞEn l,λ ðxÞ:n!mkn 1 l 1 lm 1 Theorem 10. Let n be a nonnegative integer and k ℤ. Then,lð56ÞðkÞGn,λ,p ðxÞ n!i!ð1Þi l 1,λ pðmÞλm 1 ð1Þm,1/λ ð 1Þl m!ði l 1Þmki 0 l 0il m 1 S ðl 1, mÞx 2n i βn i,λ/2: 2,λl 12ni l 1 ð59Þ
Journal of Function Spaces7From (51), we have ð kÞ Gn,λ,p ðxÞn 0tn2 u ð1 eλ ð t Þ pÞðeλ ðt Þ 1 1Þxn! eλ ðt Þ 1 k,λ2uk,λ ð1 eλ ð t Þ pÞ ðe ðt Þ 1Þi ðx Þi λe λ ðt Þ 1i!i 0 ðkÞ G n ,λ ,pn 0 n No data were used to support this study.tt ðxÞi S2,λ ðl, iÞn! i 0l!l iConflicts of Interesttn ltl ðxÞi S2,λ ðl, iÞn! i 0 i 0l!n 0!! n lntnð kÞ :ðxÞi S2,λ ðl, iÞGn l,λ,pn!n 0 l 0 i 0lðkÞð60ÞTheorem 11. Let n be a nonnegative integer and k ℤ. Then,l l 0 i 0nl!ð kÞðxÞi S2,λ ðl, iÞGn l,λ,p :The authors declare no conflict of interest.Authors’ ContributionsAll authors contributed equally to the manuscript and typed,read, and approved the final manuscript.AcknowledgmentsBy equation (60), we get the following theorem.nData Availabilityl G n ,λ ,pðkÞGn,λ,p ðxÞ degenerate versions of certain special polynomials andnumbers and their applications to physics, economics, andengineering as well as mathematics.ð61Þ4. ConclusionIn this article, we introduced degenerate poly-Genocchi polynomials and numbers by using the degenerate polylogarithmfunction and derived several properties on the degeneratepoly-Genocchi numbers. We represented the generatingfunction of the degenerate poly-Genocchi numbers by iterated integrals in Theorems 4–6 and explicit degeneratepoly-Genocchi polynomials in terms of the Euler polynomials and degenerate Stirling numbers of the second kindin Theorem 3. We also represented those numbers in termsof the degenerate Stirling numbers of the second kind in Theorem 7. In the last section, we defined the degenerateunipoly-Genocchi polynomials by using degenerate polylogarithm function and obtained the identity degenerateunipoly-Genocchi polynomials in terms of the degenerateGenocchi polynomials and degenerate Stirling numbers ofthe second kind in Theorem 8, the degenerate Euler polynomials and the degenerate Stirling numbers of the second kindin Theorem 9, the degenerate Bernoulli and degenerate Stirling numbers of the second kind in Theorem 10, and thedegenerate unipoly-Genocchi numbers and Stirling numbersof the second kind in Theorem 11. It is important that thestudy of the degenerate version is widely applied not onlyto numerical theory and combinatorial theory but also tosymmetric identity, differential equations, and probabilitytheory. In particular, many symmetric identities have beenstudied for degenerate versions of many special polynomials[1, 3, 12, 23, 29–32]. Genocchi numbers have been also extensively studied in many different branches of mathematics.The works of Genocchi numbers and their combinatorialrelations have received much attention [6–9]. With this inmind, as a future project, we would like to continue to studyThe authors would like to express the gratitude to the Deanship of Scientific Research at King Khalid University, SaudiArabia, for providing funding research group under theresearch grant number R G P.1/162/42.References[1] L. Carlitz, “Degenerate Stirling, Bernoulli and Eulerian numbers,” Utilitas Mathematica, vol. 15, pp. 51–88, 1979.[2] L. Carlitz, “A degenerate Staudt-Clausen theorem,” Archiv derMathematik, vol. 7, no. 1, pp. 28–33, 1956.[3] D. S. Kim and T. Kim, “A note on a new type of degenerateBernoulli numbers,” Russian Journal of Mathematical Physics,vol. 27, no. 2, pp. 227–235, 2020.[4] T. Kim, D. S. Kim, J. Kwon, and H. Lee, “Degenerate polyexponential functions and type 2 degenerate poly-Bernoulli numbers and polynomials,” Advances in Difference Equations,vol. 2020, no. 1, 2020.[5] T. Kim, D. S. Kim, J. Kwon, and H. Y. Kim, “A note on degenerate Genocchi and poly-Genocchi numbers and polynomials,” Journal of Inequalities and Applications, vol. 2020,no. 1, 2020.[6] T. Kim, D. S. Kim, D. V. Dolgy, and J. Kwon, “Some identitieson degenerate Genocchi and Euler numbers,” Informatica,vol. 31, no. 4, pp. 42–51, 2020.[7] H. K. Kim and L. C. Jang, “A note on degenerate polyGenocchi numbers and polynomials,” Advances in DifferenceEquations, vol. 2020, no. 1, 2020.[8] D. S. Lee and H.-Y. Kim, “On the new type of degenerate polyGenocchi numbers and polynomials,” Advances in DifferenceEquations, vol. 2020, 2020.[9] S. K. Sharma, W. A. Khan, S. Araci, and S. S. Ahmed, “Newtype of degenerate Daehee polynomials of the second kind,”Advances in Difference Equations, vol. 2020, no. 1, 2020.[10] S. K. Sharma, “A note on degenerate poly-Genocchi polynomials,” International Journal of Advanced and Applied Sciences, vol. 7, no. 5, pp. 1–5, 2020.[11] Y. Simsek, I. N. I. Cangul, V. Kurt, and D. S. Kim, “q-Genocchinumbers and polynomials associated with q-Genocchi-type lfunctions,” Advances in Difference Equations, vol. 2008, ArticleID 815750, 13 pages, 2008.
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ematics as, for instance, elementary number theory, complex analytic number theory, differential topology (differential structures on spheres), theory of modular forms (Eisenstein series), p-adic analytic number theory (p-adic L-functions), and quantum physics (quantum groups). The works of Genocchi numbers and their combinatorial relations have
Catan Family 3 4 4 Checkers Family 2 2 2 Cherry Picking Family 2 6 3 Cinco Linko Family 2 4 4 . Lost Cities Family 2 2 2 Love Letter Family 2 4 4 Machi Koro Family 2 4 4 Magic Maze Family 1 8 4 4. . Top Gun Strategy Game Family 2 4 2 Tri-Ominos Family 2 6 3,4 Trivial Pursuit: Family Edition Family 2 36 4
At T 0K (above), No occupation of states above EF and complete occupation of states below EF . Distribution. Equilibrium Carrier Concentration Formulas for n and p Degenerate vs. Non-degenerate Semiconductor Alternative Expressions for n and p ni and the np Product
described degenerate algebraic curves of genus g, obtained by choosing a number of disjoint loops and pulling them until they pop (Figure 4). Figure 4. A degenerate Riemann surface of genus 2 The result is a singular Riemann surface obtained by taking a number of usual Riemann su
what he wrote: “This effect had something to do with Lev Landau. I had a German student in Göttingen, R. Renner, and he wrote a paper on degenerate electronic states in the linear carbon dioxide molecule, assuming that the excited, degenerate state of carbon dioxide is linear. In the year 1934 bo
Th e four conic sections you have created are known as non-degenerate conic sections. A point, a line, and a pair of intersecting line are known as degenerate conics. Axis Edge Vertex Base Th e fi gures to the left illustrate a plane intersecting a double cone. Label each conic section as an ellipse, circle, parabola or hyperbola. 5. the conic .
Cross Sections - Degenerate Conic Sections The Conceptualizer! The double cone also contains other shapes that are called degenerate conic sections. If the plane goes through the vertices of the cones, the
For example, for the C3v point group: C3 Fv-1 C 3 2 F v so C3 and C3 2 belong to the same class. Elements belonging to the same class have the same characters. The number of symmetry species is equal to the number of classes, Nc 3, for both degenerate and non-degenerate point gro
at a Brillouin zone boundary, the weak perturbing potential has a very large efiect and therefore non{degenerate perturbation theory will not work in this case. For k values near a Brillouin zone boundary, we must then use degenerate perturbation theory (see AppendixA). Since the matrix elements coupling the plane wave states k and
