# A Note onq-Analogues of Degenerate Catalan-Daehee Numbers and Polynomials

Authors: Waseem A. Khan
Journal: Journal of Mathematics
 (2022)

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## Abstract

Recently, Yuankui et al. (Filomat J. 35 (5):17, 2022) studiedq-analogues of Catalan-Daehee numbers and polynomials by making use of p-adic q-integrals on ℤp. Motivated by this study, we consider q-analogues of degenerate Catalan-Daehee numbers and polynomials with the help of p-adic q-integrals on ℤp. By using their generating function, we derive some new relations including the degenerate Stirling numbers of the first and second kinds. Moreover, we also derive some new identities and properties of this type of polynomials and numbers.


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## Body

## 1. Introduction

Numerous exceptional numbers and polynomials have been concentrated by utilizing different techniques, including producing capacities,p-adic investigation, combinatorial techniques, umbral math, differential conditions, likelihood hypothesis, and scientific number hypothesis. In [1], Kuculoglu et al. developed producing capacities for new classes of Catalan-type numbers and polynomials. Utilizing these capacities and their useful conditions, they gave different personalities and relations, including these numbers and polynomials, and different classes of extraordinary numbers, polynomials, and capacities. Some endless series portrayals, including Catalan-type numbers and combinatorial numbers, were examined. In addition, a few repeat relations and computational calculations in the Python programming language represented the Catalan-type numbers and polynomials with their plots under the extraordinary circumstances. They likewise gave a few subordinate equations for these polynomials. Above, polynomials and numbers can be determined by utilizing the Riemann indispensable, shape fundamental, Volkenborn vital, and fermionic p-adic essential.Catalan-Daehee numbers and polynomials were presented in [2], and a few properties and personalities related with those numbers and polynomials were inferred by using umbral analytic procedures. The group of straight differential conditions emerging from the creating capacity of Catalan-Daehee numbers was thought of as in [3]. In [4], a few properties and personalities related with Catalan numbers and polynomials were inferred by using umbral analytic procedures. Dolgy et al. [5] gave a few new characters for those numbers and polynomials got from p-adic Volkenborn vital on ℤp. Recently, Yuankui et al. [6] presented and contemplated q-analogues of the Catalan-Daehee numbers and polynomials with the help of p-adic q-integrals on ℤp. The point of this study is to present q-analogues of the ruffian Catalan-Daehee numbers and polynomials by utilizing p-adic q-essential on ℤp and infer a few unequivocal characters for those numbers and polynomials connected with different exceptional numbers and polynomials. For the remainder of this segment, we review the important realities that are required throughout this study.Letp be a fixed odd prime number. Throughout this study, ℤp, ℚp, and ℂp will denote, respectively, the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of ℚp. The p-adic norm ⋅p is normalized as using pp=1/p. Let q be an indeterminate in ℂp with 1−qp<p−1/p−1. The q-analogue of x is defined through xq=1−qx/1−q. Note that limq⟶1xq=x. Let f be a uniformly differentiable function on ℤp. Then, the p-adic q-integral on ℤp is defined by Kim as [4, 7](1)∫ℤpfxdμqx=limN⟶∞∑x=0pN−1fxμqx+pNℤp=limN⟶∞1pNq∑x=0pN−1fxqx,see7,8.From (1), we note that(2)q∫ℤpfx+1dμqx=∫ℤpfxdμqx+q−1f0+q−1logqf′0,where f′0=df/dx|x=0 ([7–12]).Let us takefx=ext. By (1), we get(3)q−1+q−1/logqtqet−1=∫ℤpextdμqx.Theq-Bernoulli numbers are defined by ([6])(4)q−1+q−1/logqtqet−1=∑n=0∞Bn,qtnn!.From (4), we note that(5)qBq+1n−Bn,q=q−1,ifn=0,q−1logq,ifn=1,0,ifn>1,with the usual convention about replacing Bqn by Bn,q.The Catalan numbers are defined by the generating function as follows ([1, 4, 6, 13, 14]):(6)21+1−4t=1−1−4t2t=∑n=0∞Cntn,where t∈ℂp with tp<p−1/p−1 and Cn=2nn1/n+1,n≥0.The Catalan polynomials are defined by the generating function as follows ([4, 13]):(7)∫ℤp1−4tx+y/2dμ−1y=21+1−4t1−4tx/2=∑n=0∞Cnxtn,where t∈ℂp with tp<p−1/p−1.Whenx=0, Cn=Cn0 are called the Catalan numbers.Thus, by (6) and (7), we have(8)Cnx=∑m=0n∑j=0mx2jS1m,j−4mCn−mm!.Kim-Kim [2, 3] introduced the Catalan-Daehee polynomials defined by(9)∫ℤp1−4tx+y/2dμy=1/2log1−4t1−4t−11−4tx/2=∑n=0∞dnxtn.Whenx=0, dn=dn0 are called the Catalan-Daehee numbers.From (6) and (9), we get(10)dn=1,ifn=0,4nn+1−∑m=0n−14n−m−1n−mCm,ifn≥1.By using (2), the q-analogues of Catalan-Daehee numbers are defined by ([6])(11)∫ℤp1−4tx/2dμqx=q−1+q−1/logq1/2log1−4tq1−4t−1=∑n=0∞dn,qtn.Note thatlimq⟶1dn,q=dn,n≥0.Jeong et al. [15] introduced the degenerate q-Daehee polynomials defined by(12)∫ℤp1+log1+λt1/λx+ydμqy=q−1+q−1/logqlog1+log1+λt1/λq−1+qlog1+λt1/λ·1+log1+λt1/λx=∑n=0∞Dn,qx|λtnn!.In the case whenx=0, Dn,qλ=Dn,q0|λ are called the degenerate q-Daehee numbers.Note that(13)limλ⟶0Dn,qx|λ=Dn,qx,n≥0.Forn≥0, the Stirling numbers of the first kind are defined by ([4, 5, 8, 9, 13, 15, 16])(14)xn=∑l=0nS1n,lxl,where x0=1, and xn=xx−1⋯x−n+1,n≥1. From (14), it is easy to see that ([1–3, 7, 10, 14, 17, 18])(15)1l!log1+tl=∑n=l∞S1n,ltnn!.Forn≥0, the Stirling numbers of the second kind are defined by ([6, 11, 12, 18–22])(16)xn=∑l=0nS2n,lxl.From (16), we see that(17)1l!et−1l=∑n=l∞S2n,ltnn!.


## 2. Theq-Analogues of Degenerate Catalan-Daehee Numbers and Polynomials

In this section, we introduceq-analogues of degenerate Catalan-Daehee numbers which are derived from the fermionic p-adic q-integral on ℤp. First, we present the following definition.Forλ,t,q∈ℂp with 1−qp<1 and λt<p−1/p−1, 1+λtx/λ=ex/λlog1+λt. Now, we define the q-analogue of degenerate Catalan-Daehee numbers which are given by the generating function(18)∫ℤp1−4log1+λt1/λx/2dμqx=q−1+q−1/logq1/2log1−4log1+λt1/λq1−4log1+λt1/λ−1=∑n=0∞dn,λ,qtn.Note that [13](19)limλ⟶0dn,λ,q=dn,q,n≥0see6.From (6) and (18), we have(20)∑n=0∞dn,λ,qtn=122q−1+q−1/logqlog1−4log1+λt1/λq21−4log1+λt1/λ−1q1−4log1+λt1/λ+1=12∑l=0∞−4lDl,q0|λtll!1+q−2q∑m=0∞Cm,λtm+1=2q2∑n=0∞−4nDn,q0|λn!tn−q∑n=1∞∑m=0n−1−4n−m−1n−m−1!Dn−m−1,q0|λCm,λtn=1+∑n=1∞2q2−4nn!Dn,q0|λtn−q∑n=1∞∑m=0n−1−4n−m−1n−m−1!Dn−m−1,q0|λCm,λtn=1+∑n=1∞2q2−4nn!Dn,q0|λ−q∑m=0n−1−4n−m−1n−m−1!Dn−m−1,q0|λCm,λtn.Therefore, by comparing the coefficients on both sides of (20), we obtain the following theorem.Theorem 1.
Forn≥0, we have(21)dn,λ,q=1,ifn=0,2q2−4nn!Dn,q0|λ−q∑m=0n−1−4n−m−1n−m−1!Dn−m−1,q0|λCm,λ,ifn≥1.By (18), we note that(22)∫ℤp1−4log1+λt1/λx/2dμqx=∑m=0∞∫ℤpxmdμqx12m1m!−4log1+λt1/λm=∑m=0∞Bm,q−2mλn−m∑n=m∞S1n,mtnn!=∑n=0∞∑m=0nBm,q−2mλn−mS1n,mtnn!.Therefore, by (18) and (22), we obtain the following theorem.Theorem 2.
Forn≥0, we have(23)dn,λ,q=−1mn!∑m=0nBm,q2mλn−mS1n,m.On replacingt by 1/λeλt−1 in (18), we have(24)∫ℤp1−4tx/2dμqx=∑m=0∞dm,λ,qm!1/λeλt−1mm!=∑m=0∞dm,λ,qλ−mm!∑n=m∞S2n,mλntnn!=∑n=0∞∑m=0ndm,λ,qλn−mm!S2n,mtnn!.On the other hand,(25)∫ℤp1−4tx/2dμqx=q−1+q−1/logq1/2log1−4tq1−4t−1=∑n=0∞dn,qtn.Therefore, by (24) and (25), we obtain the following theorem.Theorem 3.
Forn≥0, we have(26)dn,q=∑m=0ndm,λ,qλn−mS2n,mm!n!,∑m=0n−1n22n−mBm,qS1n,m=∑m=0ndm,λ,qλn−mS2n,mm!.From (18), we observe that(27)∫ℤp1−4log1+λt1/λx/2dμqx=∑m=0∞−1m4m∫ℤpx2mdμqxlog1+λt1/λmm!=∑m=0∞dm,q−1m4mλn−m∑n=m∞S1n,mtnn!=∑n=0∞∑m=0ndm,q−1m4mλn−mS1n,mtnn!.Therefore, by (18) and (27), we get the following theorem.Theorem 4.
Forn≥0, we have(28)dn,λ,q=1n!∑m=0ndm−1m4mλn−mS1n,m.Forλ,t,q∈ℂp with 1−qp<1 and λt<p−1/p−1. The q-analogue of λ-Daehee polynomials Dn,λ,qx are defined by the following generating function ([6]):(29)∫ℤp1+tλy+xdμqy=q−1+λq−1/logqlog1+tq1+tλ−11+tx=∑n=0∞Dn,λ,qxtnn!.Whenx=0, Dn,λ,q=Dn,λ,q0 are called the q-analogue of λ-Daehee numbers.On settingλ=1/2 and t⟶−4log1+λt1/λ in (29), we have(30)∫ℤp1−4log1+λt1/λx/2dμqx=∑m=0∞Dm,1/2,q−4log1+λt1/λmm!=∑m=0∞Dm,1/2,q−4mλ−m∑n=m∞S1n,mλntnn!=∑n=0∞∑m=0nDm,1/2,qλn−mS1n,m−4mn!tn.Therefore, by (18) and (30), we obtain the following theorem.Theorem 5.
Forn≥0, we have(31)dn,λ,q=∑m=0nDm,1/2,qλn−mS1n,m−4mn!,∑m=0n−1mBm,q2mλn−mS1n,m=∑m=0nDm,1/2,qλn−mS1n,m−4m.By replacingt by log1+λt1/λ in (11), we get(32)q−1+q−1/logq1/2log1−4log1+λt1/λq1−4log1+λt1/λ−1=∑m=0∞dm,qm!log1+λt1/λmm!=∑m=0∞dm,qλ−mm!log1+λtmm!=∑m=0∞dm,qλ−mm!∑n=m∞S1n,mλntnn!=∑n=0∞∑m=0ndm,qλn−mm!S1n,mtnn!.Therefore, by (18) and (32), we get the following theorem.Theorem 6.
Forn≥0, we have(33)dn,λ,q=∑m=0ndm,qλn−mm!S1n,mm!n!.Now, we observe that(34)1−4log1+λt1/λx/2=∑m=0∞x2m−1m4mlog1+λt1/λmm!=∑m=0∞x2mλ−m−1m4m∑n=m∞S1n,mλntnn!=∑n=0∞∑m=0nx2mλn−m−1m4mS1n,mtnn!.Now, we consider theq-analogue of degenerate Catalan-Daehee polynomials which are given by the following generating function:(35)∫ℤp1−4log1+λt1/λx+y/2dμqy=q−1+q−1/logq1/2log1−4log1+λt1/λq1−4log1+λt1/λ−11−4log1+λt1/λx/2=∑n=0∞dn,λ,qxtn.Whenx=0, dn,λ,q=dn,λ,q0 are called the q-analogue of degenerate Catalan-Daehee numbers.From (35), we note that(36)q−1+q−1/logq1/2log1−4log1+λt1/λq1−4log1+λt1/λ−11−4log1+λt1/λx/2=∑n=0∞dn,λ,qtn∑m=0∞x2m−1m22mm!log1+λt1/λmm!=∑n=0∞dn,λ,qtn∑m=0∞x2m−1m22mλ−mm!∑l=m∞S1l,mλltll!=∑n=0∞dn,λ,qtn∑l=0∞∑m=0lx2m−1m22mλl−mm!S1l,mtll!=∑n=0∞∑l=0n∑m=0lx2m−1m22mλl−mS1l,mdn−l,λ,qm!l!tn.By (35) and (36), we obtain the following theorem.Theorem 7.
Forn≥0, we have(37)dn,λ,qx=∑l=0n∑m=0lx2m−1m22mλl−mS1l,mdn−l,λ,qm!l!.From (35), we see that(38)∑n=0∞dn,λ,qxtn=q−1+q−1/logq1/2log1−4log1+λt1/λq1−4log1+λt1/λ−11−4log1+λt1/λx/2=∑n=0∞dn,λ,qtn∑k=0∞∑m=0kx2mλk−m−1m4mS1k,mtkk!=∑n=0∞∑k=0n∑m=0kx2mλk−mS1k,m−1m4mdn−k,λ,q1k!tn.Therefore, by (38), we obtain the following theorem.Theorem 8.
Forn≥0, we have(39)dn,q,λx=∑k=0n∑m=0kx2mλk−m−1m4mS1k,mdn−k,λ,q1k!.On replacingt by 1/λeλt−1 in (35), we define q-analogue Catalan polynomials are given by(40)∫ℤp1−4tx+y/2dμqy=q−1+q−1/logq1/2log1−4tq1−4t−11−4tx=∑n=0∞dn,qxtn.On the other hand,(41)∑m=0∞dm,λ,qxm!1/λeλt−1mm!=∑m=0∞dm,λ,qxm!λ−m∑n=m∞S1n,mλntnn!=∑n=0∞∑m=0ndm,λ,qxλn−mS1n,mm!tnn!.Therefore, by (40) and (41), we state the following theorem.Theorem 9.
Forn≥0, we have(42)dn,qx=∑m=0ndm,λ,qxλn−mS1n,mm!n!.From (35), we have(43)q−1+q−1/logq1/2log1−4log1+λt1/λq1−4log1+λt1/λ−11−4log1+λt1/λx2=∫ℤp1−4log1+λt1/λx+y/2dμqy=∑m=0∞2−m1m!−4log1+λt1/λm∫ℤpx+ymdμqy=∑m=0∞2m−1mBm,qxλ−m∑n=m∞S1n,mλntnn!=∑n=0∞∑m=0n2m−1mBm,qxλn−mS1n,m1n!tn.Thus, by (35) and (43), we get the following theorem.Theorem 10.
Forn≥0, we have(44)dn,λ,qx=∑m=0n2m−1mBmxλn−mS1n,m1n!.Replacingt by e−λt/4−1/λ in (35), we get(45)∫ℤp1+tx+y/2dμqy=∑n=0∞2−n∫ℤpx+yndμqytnn!.On the other hand, we have(46)∑m=0∞dm,λ,qxm!e−λt/4−1/λmm!=∑m=0∞dm,q,λxm!λ−m∑n=m∞S2n,m−4nλntnn!=∑n=0∞∑m=0ndm,λ,qxm!λn−m−1n4nS2n,mtnn!.Therefore, by (45) and (46), we obtain the following theorem.Theorem 11.
Forn≥0, we have(47)∫ℤpx+yndμqy=−2n∑m=0ndm,λ,qxm!λn−mS2n,m.


## 3. Conclusion

By utilizing various apparatuses, many unique numbers and polynomials have been concentrated. Beforehand, the Catalan-Daehee numbers and polynomials were presented throughp-adic Volkenborn, and a few intriguing outcomes for them were obtained by utilizing creating capacities, differential conditions, umbral math, and p-adic Volkenborn integrals. In this study, we presented degenerate q-analogues of the Catalan-Daehee numbers and polynomials and acquired a few unequivocal articulations and personalities connected with them. In more detail, we communicated the Catalan-Daehee numbers as far as ruffian q-Daehee numbers and the q-Bernoulli polynomials and Stirling quantities of the primary kind. We acquired a personality, including q-Bernoulli numbers, degenerate q-Catalan-Daehee numbers, and Stirling quantities of the subsequent kind. Likewise, we got an unequivocal articulation for the ruffian q-analogues of Catalan-Daehee polynomials, which include the savage q-analogues of Catalan-Daehee numbers and Stirling quantities of the primary kind.



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