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Some identities on degenerate hyperbolic functions arising from p-adic integrals on Zp

  • The aim of this paper is to introduce several degenerate hyperbolic functions as degenerate versions of the hyperbolic functions, to evaluate Volkenborn and the fermionic p-adic integrals of the degenerate hyperbolic cosine and the degenerate hyperbolic sine functions and to derive from them some identities involving the degenerate Bernoulli numbers, the degenerate Euler numbers and the Cauchy numbers of the first kind.

    Citation: Taekyun Kim, Hye Kyung Kim, Dae San Kim. Some identities on degenerate hyperbolic functions arising from p-adic integrals on Zp[J]. AIMS Mathematics, 2023, 8(11): 25443-25453. doi: 10.3934/math.20231298

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  • The aim of this paper is to introduce several degenerate hyperbolic functions as degenerate versions of the hyperbolic functions, to evaluate Volkenborn and the fermionic p-adic integrals of the degenerate hyperbolic cosine and the degenerate hyperbolic sine functions and to derive from them some identities involving the degenerate Bernoulli numbers, the degenerate Euler numbers and the Cauchy numbers of the first kind.



    Volkenborn and the fermionic p-adic integrals of the powers of x give respectively the Bernoulli numbers and the Euler numbers, while Volkenborn and the fermionic p-adic integrals of the generalized falling factorials yield respectively the degenerate Bernoulli numbers and the degenerate Euler numbers (see (1.12)). Thus the latter may be viewed as degenerate versions of the former.

    The aim of this paper is to derive some identities on degenerate hyperbolic functions arising from Volkenborn and the fermionic p-adic integrals on Zp. In more detail, we introduce degenerate hyperbolic functions as natural degenerate versions of the usual hyperbolic functions and derive some identities relating to them. Then we evaluate Volkenborn and the fermionic p-adic integrals of the degenerate hyperbolic cosine and the degenerate hyperbolic sine functions. From those results, we derive some identities involving the degenerate Bernoulli numbers, the degenerate Euler numbers and the Cauchy numbers of the first kind.

    The outline of this paper is as follows. In Section 1, we recall Volkenborn integral of uniformly differentiable functions on Zp and the fermionic p-adic integral of continuous functions on Zp, together with their integral equations and examples in the case of exponential functions. We remind the reader of Cauchy numbers of the first kind. We recall the degenerate exponentials as a degenerate version of the usual exponentials. Then we show that the degenerate Bernoulli and the degenerate Euler numbers arise naturally respectively from the Volkenborn and the fermionic p-adic integrals on Zp of the degenerate exponentials. Section 2 is the main result of this paper. We introduce degenerate versions of the hyperbolic functions, namely the degenerate hyperbolic cosine coshλ(x:a), the degenerate hyperbolic sine sinhλ(x:a), degenerate hyperbolic tangent tanhλ(x:a) and the degenerate hyperbolic cotangent cothλ(x:a). Then we derive several identities connecting those degenerate hyperbolic functions. We evaluate Volkenborn integrals of coshλ(x:a) and sinhλ(x:a). From those results, we obtain some identities involving the degenerate Bernoulli numbers and the Cauchy numbers of the first kind. We compute the fermionic p-adic integrals of the same degenerate hyperbolic functions, from which we derive an identity involving the degenerate Euler numbers. Finally, we conclude this paper in Section 3. In the rest of this section, we recall the necessary facts that are needed throughout this paper.

    Let p be a fixed odd prime number. Throughout this paper, Zp,Qp and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of an algebraic closure of Qp, respectively. The p-adic norm ||p is normalized as |p|p=1p. The Volkenborn integral on Zp is defined by

    Zpf(x) dμ(x)=limNPN1x=0f(x)μ(x+pNZp)=limN1pNpN1x=0f(x),(see [1–7]). (1.1)

    From (1.1), we note that

    Zpf(x+1) dμ(x)Zpf(x) dμ(x)=f(0),(see [3,7]), (1.2)

    where f is a Cp-valued uniform differentiable function on Zp.

    In [5], the fermionic p-adic integral on Zp is defined by

    Zpf(x) dμ1(x)=limNpN1x=0f(x) μ1(x+pNZp)=limNpN1x=0f(x)(1)x, (1.3)

    where f is a Cp-valued continuous function on Zp.

    Thus, by (1.3), we get

    Zpf(x+1) dμ1(x)+Zpf(x) dμ1(x)=2f(0)(see [1,5,6]). (1.4)

    Adopting the notation from [7], we let E denote the additive group given by

    E={xCp||x|p<p1p1}. (1.5)

    It is known that there is an isomorphism from the multiplicative group 1+E to the additive group E given by 1+xlog(1+x), where log=logp is the p-adic logarithm. Moreover, the inverse map is given by xex=expx, where exp=expp is the p-adic exponential (see [7]). We note that

    |1xlog(1+x)|p1,forxE. (1.6)

    The Cauchy numbers of the first kind (also called Bernoulli numbers of the second) are defined by

    tlog(1+t)=n=0Cntnn!,(|t|p<1,t0).

    From (1.2) and (1.4), we note that

    Zpext dμ(x)=tet1=n=0Bntnn!,(tE,t0),(see [1–18]), (1.7)

    and

    Zpext dμ1(x)=2et+1=n=0Entnn!,(tE),(see [1,3,5]), (1.8)

    where Bn and En are the n-th Bernoulli number and the n-th Euler number, respectively.

    Throughout this paper, we let λ be any nonzero element in Cp.

    The degenerate exponentials are defined by

    exλ(t)=n=0(x)n,λtnn!=(1+λt)xλ,(t1λE1xE),(see [8,9]), (1.9)

    where the generalized falling factorials (also called the λ-falling factorials) are given by (x)0,λ=1,(x)n,λ=x(xλ)(x(n1)λ),(n1).

    From (1.6), we see that xtλtlog(1+λt) is in the region of convergence E of ex, when λtE and xtE. Thus exλ(t)=exp(xtλtlog(1+λt)) is convergent when t1λE1xE. Here we understand 1λE1xE=1λE, for x=0. Especially, eλ(t)=e1λ(t) is convergent for t1λEE.

    From (1.2), we observe that

    Zpexλ(t) dμ(x)=1λlog(1+λt)eλ(t)1=n=0βn,λtnn!,(t1λEE), (1.10)

    where βn,λ are called the (fully) degenerate Bernoulli numbers (see [3,10]) .

    Note that limλ0βn,λ=Bn, (n0), (see [11]) .

    From (1.4), we note that

    Zpexλ(t) dμ1(x)=2eλ(t)+1=n=0En,λtnn!,(t1λEE), (1.11)

    where En,λ are called the degenerate Euler numbers (see [3,10]) .

    Note that limλ0En,λ=En, (n0).

    Thus, by (1.10) and (1.11), we get

    Zp(x)n,λ dμ(x)=βn,λ,  Zp(x)n,λ dμ1(x)=En,λ, (n0). (1.12)

    The hyperbolic functions of real or complex variables are given by

    coshx=ex+ex2,sinhx=exex2,tanhx=sinhxcoshx,andcothx=coshxsinhx. (1.13)

    Let λ be any nonzero element in Cp, and let a1λE1xE, (see (1.5)). Let us consider the degenerate hyperbolic functions given by

    coshλ(x:a)=12(exλ(a)+exλ(a)),sinhλ(x:a)=12(exλ(a)exλ(a)),tanhλ(x:a)=sinhλ(x:a)coshλ(x:a),cothλ(x:a)=coshλ(x:a)sinhλ(x:a),(x0,a0). (2.1)

    Note that limλ0coshλ(x:a)=cosh(xa),  limλ0sinhλ(x:a)=sinh(xa), limλ0tanhλ(x:a)=tanh(xa),  and  limλ0cothλ(x:a)=coth(xa).

    From (2.1), we note that

    coshλ(2x:a)=e2xλ(a)+e2xλ(a)2=1+e2xλ(a)+e2xλ(a)22=2(exλ(a)exλ(a)2)2+1=1+2sinh2λ(x:a). (2.2)

    On the other hand, by (2.1), we get

    2cosh2λ(x:a)1=2×e2xλ(a)+e2xλ(a)+241=e2xλ(a)+e2xλ(a)2=coshλ(2x:a). (2.3)

    Therefore, by (2.2) and (2.3), we obtain the following proposition.

    Proposition 1. Let a1λE1xE. Then the following identity holds true.

    coshλ(2x:a)=2cosh2λ(x:a)1=1+2sinh2λ(x:a).

    By (2.1), we get

    sinhλ(2x:a)=e2xλ(a)e2xλ(a)2=2exλ(a)exλ(a)2exλ(a)+exλ(a)2=2sinhλ(x:a)coshλ(x:a). (2.4)

    Thus we have the following proposition.

    Proposition 2. Let a1λE1xE. Then the following identity holds true.

    sinhλ(2x:a)=2sinhλ(x:a)coshλ(x:a).

    Now, we observe that

    sinhλ(x+y:a)=12(ex+yλ(a)exyλ(a))=ex+yλ(a)exyλ(a)+exyλ(a)ex+yλ(a)4+ex+yλ(a)exyλ(a)exyλ(a)+ex+yλ(a)4=exλ(a)exλ(a)2×eyλ(a)+eyλ(a)2+exλ(a)+exλ(a)2×eyλ(a)eyλ(a)2=sinhλ(x:a)coshλ(y:a)+coshλ(x:a)sinhλ(y:a),(a1λE1xE1yE). (2.5)

    On the other hand, by (2.1), we get

    coshλ(x:a)coshλ(y:a)+sinhλ(x:a)sinhλ(y:a)=exλ(a)+exλ(a)2×eyλ(a)+eyλ(a)2+exλ(a)exλ(a)2×eyλ(a)eyλ(a)2=ex+yλ(a)+exyλ(a)+ex+yλ(a)+exyλ(a)4+ex+yλ(a)+exyλ(a)exyλ(a)ex+yλ(a)4=2(ex+yλ(a)+e(x+y)λ(a))4=ex+yλ(a)+e(x+y)λ(a)2=coshλ(x+y:a), (2.6)

    where a1λE1xE1yE.

    Therefore, by (2.5) and (2.6), we obtain the following proposition.

    Proposition 3. Let a1λE1xE1yE. Then the following identities are valid.

    coshλ(x+y:a)=coshλ(x:a)coshλ(y:a)+sinhλ(x:a)sinhλ(y:a),andsinhλ(x+y:a)=sinhλ(x:a)coshλ(y:a)+coshλ(x:a)sinhλ(y:a).

    Noting that ddxcoshλ(x:a)=1λlog(1+λa)sinhλ(x:a) and using (1.2), we have

    0=Zpcoshλ(x+1:a) dμ(x)Zpcoshλ(x:a) dμ(x)=(coshλ(1:a)1)Zpcoshλ(x:a) dμ(x)+sinhλ(1:a)Zpsinhλ(x:a) dμ(x), (2.7)

    where a1λEE.

    Observing that ddxsinhλ(x:a)=1λlog(1+λa)coshλ(x:a) and making use of (1.2), we get

    1λlog(1+λa)=Zpsinhλ(x+1:a)dμ(x)Zpsinhλ(x:a)dμ(x)=(coshλ(1:a)1)Zpsinhλ(x:a) dμ(x)+sinhλ(1:a)Zpcoshλ(x:a) dμ(x), (2.8)

    where a1λEE.

    By solving the system of linear equations in (2.7) and (2.8), we get

    Zpcoshλ(x:a) dμ(x)=1λlog(1+λa)sinhλ(1:a)2(coshλ(1:a)1),Zpsinhλ(x:a) dμ(x)=121λlog(1+λa),(a1λEE). (2.9)

    From (2.9) and Propositions 1 and 2, we have

    Zpcoshλ(x:a)dμ(x)=1λlog(1+λa)2sinhλ(12:a)coshλ(12:a)4sinh2λ(12:a)=a2cothλ(12:a)log(1+λa)λa,(a1λEE). (2.10)

    Therefore, from (2.9) and (2.10) we obtain the following theorem.

    Theorem 4. Let a1λEE. Then the following relations hold true.

    λalog(1+λa)Zpcoshλ(x:a) dμ(x)=a2cothλ(12:a),λalog(1+λa)Zpsinhλ(x:a) dμ(x)=a2.

    By Taylor expansion, we get

    coshλ(x:a)=12(exλ(a)+exλ(a))=12(exλ(a)+exλ(a))=12m=0((x)m,λ+(1)m(x)m,λ)amm!,(a1λE1xE). (2.11)

    From Theorem 4, (2.11) and (1.12), we have

    a2cothλ(12:a)=λalog(1+λa)Zpcoshλ(x:a) dμ(x)=(l=0λlClall!)12m=0Zp((x)m,λ+(1)m(x)m,λ)dμ(x)amm!=l=0λlClall!m=0(βm,λ+(1)mβm,λ2)amm!=n=0(nm=0(nm)(βm,λ+(1)mβm,λ2)Cnmλnm)ann!, (2.12)

    where a1λEE. By Taylor expansion, we get

    sinhλ(x:a)=12m=0((x)m,λ(1)m(x)m,λ)amm!,(a1λE1xE). (2.13)

    Thus, by (2.9) and (2.13), we get

    12λlog(1+λa)=Zpsinhλ(x:a) dμ(x)=n=0(βn,λ(1)nβn,λ2)ann!=n=1(βn,λ(1)nβn,λ2)ann!, (2.14)

    (a1λEE),

    and

    12λlog(1+λa)=12n=1(1)n1λn1nan=12n=1(n1)!(1)n1λn1ann!,(|λa|p1). (2.15)

    By (2.12), (2.14) and (2.15), we obtain the following theorem.

    Theorem 5. For nN, we have the identity:

    (n1)!(λ)n1=βn,λ(1)nβn,λ.

    In addition, we have the following relation:

    a2cothλ(12:a)=n=0(nm=0(nm)(βm,λ+(1)mβm,λ2)Cnmλnm)ann!,

    where a1λEE.

    From (1.4), we note that

    2=Zpcoshλ(x+1:a) dμ1(x)+Zpcoshλ(x:a) dμ1(x)=(coshλ(1:a)+1)Zpcoshλ(x:a) dμ1(x)+sinhλ(1:a)Zpsinhλ(x:a) dμ1(x), (2.16)

    and

    0=Zpsinhλ(x+1:a) dμ1(x)+Zpsinhλ(x:a) dμ1(x)=(coshλ(1:a)+1)Zpsinhλ(x:a) dμ1(x)+sinhλ(1:a)Zpcoshλ(x:a) dμ1(x), (2.17)

    where a1λEE.

    By solving the system of linear equations in (2.16) and (2.17), we obtain

    Zpcoshλ(x:a) dμ1(x)=1,Zpsinhλ(x:a) dμ1(x)=sinhλ(1:a)coshλ(1:a)+1, (2.18)

    where a1λEE.

    Thus, by (2.18) and Propositions 1 and 2, we get

    Zpsinhλ(x:a) dμ1(x)=2sinhλ(12:a)coshλ(12:a)2cosh2λ(12:a)=tanhλ(12:a), (2.19)

    where a1λEE.

    Therefore, by (2.18) and (2.19), we obtain the following theorem.

    Theorem 6. Let a1λEE. Then we have the following relations.

    Zpsinhλ(x:a) dμ1(x)=tanhλ(12:a),Zpcoshλ(x:a) dμ1(x)=1.

    Thus, by Theorem 6, (2.13) and (1.12), we get

    tanhλ(12:a)=Zpsinhλ(x:a) dμ1(x)=n=0(En,λ(1)nEn,λ2)ann!,(a1λEE). (2.20)

    In addition, by Theorem 6, (2.11) and (1.12), we get

    1=Zpcoshλ(x;a) dμ1(x)=n=0(En,λ+(1)nEn,λ2)ann!,(a1λEE). (2.21)

    Thus, by(2.21), we get

    12(En,λ+(1)nEn,λ)={1,ifn=0,0,ifn>0. (2.22)

    For n1, we have

    En,λ=(1)n1En,λ. (2.23)

    From (2.20) and (2.23), we obtain the following theorem.

    Theorem 7. Let a1λEE. Then we have the following relation.

    tanhλ(12:a)=n=1En,λann!.

    We introduced several degenerate hyperbolic functions which are degenerate versions of the usual hyperbolic functions. We computed Volkenborn and the fermionic p-adic integrals of the degenerate hyperbolic cosine and the degenerate hyperbolic sine functions. From those results, we were able to derive some identities regarding the degenerate hyperbolic tangent and the degenerate hyperbolic cotangent functions.

    In recent years, various kinds of tools, like generating functions, combinatorial methods, p-adic analysis, umbral calculus, differential equations, probability theory, special functions, analytic number theory and operator theory, have been used in studying special numbers and polynomials, and degenerate versions of them.

    It is one of our future research projects to continue to explore many special numbers and polynomials and their applications to physics, science and engineering as well as to mathematics.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors would like to thank the reviewers for their valuable comments and suggestions. All authors thank Jangjeon Institute for Mathematical Science for the support of this research. The second author is supported by the Basic Science Research Program, the National Research Foundation of Korea, (NRF-2021R1F1A1050151).

    Taekyun Kim, Hye Kyung Kim and Dae Sa Kim are the Guest Editors of special issue "Number theory, combinatorics and their applications: theory and computation" for AIMS Mathematics. Taekyun Kim, Hye Kyung Kim and Dae Sa Kim were not involved in the editorial review and the decision to publish this article.



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