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)=limN→∞PN−1∑x=0f(x)μ(x+pNZp)=limN→∞1pNpN−1∑x=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) dμ−1(x)=limN→∞pN−1∑x=0f(x) μ−1(x+pNZp)=limN→∞pN−1∑x=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) dμ−1(x)+∫Zpf(x) dμ−1(x)=2f(0)(see [1,5,6]). | (1.4) |
Adopting the notation from [7], we let E denote the additive group given by
E={x∈Cp||x|p<p−1p−1}. | (1.5) |
It is known that there is an isomorphism from the multiplicative group 1+E to the additive group E given by 1+x↦log(1+x), where log=logp is the p-adic logarithm. Moreover, the inverse map is given by x↦ex=expx, where exp=expp is the p-adic exponential (see [7]). We note that
|1xlog(1+x)|p≤1,forx∈E. | (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,t≠0). |
From (1.2) and (1.4), we note that
∫Zpext dμ(x)=tet−1=∞∑n=0Bntnn!,(t∈E,t≠0),(see [1–18]), | (1.7) |
and
∫Zpext dμ−1(x)=2et+1=∞∑n=0Entnn!,(t∈E),(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,(t∈1λE∩1xE),(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(x−λ)⋯(x−(n−1)λ),(n≥1).
From (1.6), we see that xtλtlog(1+λt) is in the region of convergence E of ex, when λt∈E and xt∈E. Thus exλ(t)=exp(xtλtlog(1+λt)) is convergent when t∈1λE∩1xE. Here we understand 1λE∩1xE=1λE, for x=0. Especially, eλ(t)=e1λ(t) is convergent for t∈1λE∩E.
From (1.2), we observe that
∫Zpexλ(t) dμ(x)=1λlog(1+λt)eλ(t)−1=∞∑n=0βn,λtnn!,(t∈1λE∩E), | (1.10) |
where βn,λ are called the (fully) degenerate Bernoulli numbers (see [3,10]) .
Note that limλ→0βn,λ=Bn, (n≥0), (see [11]) .
From (1.4), we note that
∫Zpexλ(t) dμ−1(x)=2eλ(t)+1=∞∑n=0En,λtnn!,(t∈1λE∩E), | (1.11) |
where En,λ are called the degenerate Euler numbers (see [3,10]) .
Note that limλ→0En,λ=En, (n≥0).
Thus, by (1.10) and (1.11), we get
∫Zp(x)n,λ dμ(x)=βn,λ, ∫Zp(x)n,λ dμ−1(x)=En,λ, (n≥0). | (1.12) |
The hyperbolic functions of real or complex variables are given by
coshx=ex+e−x2,sinhx=ex−e−x2,tanhx=sinhxcoshx,andcothx=coshxsinhx. | (1.13) |
Let λ be any nonzero element in Cp, and let a∈1λE∩1xE, (see (1.5)). Let us consider the degenerate hyperbolic functions given by
coshλ(x:a)=12(exλ(a)+e−xλ(a)),sinhλ(x:a)=12(exλ(a)−e−xλ(a)),tanhλ(x:a)=sinhλ(x:a)coshλ(x:a),cothλ(x:a)=coshλ(x:a)sinhλ(x:a),(x≠0,a≠0). | (2.1) |
Note that limλ→0coshλ(x:a)=cosh(xa), limλ→0sinhλ(x:a)=sinh(xa), limλ→0tanhλ(x:a)=tanh(xa), and limλ→0cothλ(x:a)=coth(xa).
From (2.1), we note that
coshλ(2x:a)=e2xλ(a)+e−2xλ(a)2=1+e2xλ(a)+e−2xλ(a)−22=2(exλ(a)−e−xλ(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)+e−2xλ(a)+24−1=e2xλ(a)+e−2xλ(a)2=coshλ(2x:a). | (2.3) |
Therefore, by (2.2) and (2.3), we obtain the following proposition.
Proposition 1. Let a∈1λE∩1xE. 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)−e−2xλ(a)2=2exλ(a)−e−xλ(a)2exλ(a)+e−xλ(a)2=2sinhλ(x:a)coshλ(x:a). | (2.4) |
Thus we have the following proposition.
Proposition 2. Let a∈1λE∩1xE. 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)−e−x−yλ(a))=ex+yλ(a)−e−x−yλ(a)+ex−yλ(a)−e−x+yλ(a)4+ex+yλ(a)−e−x−yλ(a)−ex−yλ(a)+e−x+yλ(a)4=exλ(a)−e−xλ(a)2×eyλ(a)+e−yλ(a)2+exλ(a)+e−xλ(a)2×eyλ(a)−e−yλ(a)2=sinhλ(x:a)coshλ(y:a)+coshλ(x:a)sinhλ(y:a),(a∈1λE∩1xE∩1yE). | (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)+e−xλ(a)2×eyλ(a)+e−yλ(a)2+exλ(a)−e−xλ(a)2×eyλ(a)−e−yλ(a)2=ex+yλ(a)+e−x−yλ(a)+e−x+yλ(a)+ex−yλ(a)4+ex+yλ(a)+e−x−yλ(a)−ex−yλ(a)−e−x+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 a∈1λE∩1xE∩1yE.
Therefore, by (2.5) and (2.6), we obtain the following proposition.
Proposition 3. Let a∈1λE∩1xE∩1yE. 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 a∈1λE∩E.
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 a∈1λE∩E.
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),(a∈1λE∩E). | (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,(a∈1λE∩E). | (2.10) |
Therefore, from (2.9) and (2.10) we obtain the following theorem.
Theorem 4. Let a∈1λE∩E. 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)+e−xλ(a))=12(exλ(a)+ex−λ(−a))=12∞∑m=0((x)m,λ+(−1)m(x)m,−λ)amm!,(a∈1λE∩1xE). | (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!)12∞∑m=0∫Zp((x)m,λ+(−1)m(x)m,−λ)dμ(x)amm!=∞∑l=0λlClall!∞∑m=0(βm,λ+(−1)mβm,−λ2)amm!=∞∑n=0(n∑m=0(nm)(βm,λ+(−1)mβm,−λ2)Cn−mλn−m)ann!, | (2.12) |
where a∈1λE∩E. By Taylor expansion, we get
sinhλ(x:a)=12∞∑m=0((x)m,λ−(−1)m(x)m,−λ)amm!,(a∈1λE∩1xE). | (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) |
(a∈1λE∩E),
and
−12λlog(1+λa)=−12∞∑n=1(−1)n−1λn−1nan=−12∞∑n=1(n−1)!(−1)n−1λn−1ann!,(|λa|p≤1). | (2.15) |
By (2.12), (2.14) and (2.15), we obtain the following theorem.
Theorem 5. For n∈N, we have the identity:
−(n−1)!(−λ)n−1=βn,λ−(−1)nβn,−λ. |
In addition, we have the following relation:
a2cothλ(12:a)=∞∑n=0(n∑m=0(nm)(βm,λ+(−1)mβm,−λ2)Cn−mλn−m)ann!, |
where a∈1λE∩E.
From (1.4), we note that
2=∫Zpcoshλ(x+1:a) dμ−1(x)+∫Zpcoshλ(x:a) dμ−1(x)=(coshλ(1:a)+1)∫Zpcoshλ(x:a) dμ−1(x)+sinhλ(1:a)∫Zpsinhλ(x:a) dμ−1(x), | (2.16) |
and
0=∫Zpsinhλ(x+1:a) dμ−1(x)+∫Zpsinhλ(x:a) dμ−1(x)=(coshλ(1:a)+1)∫Zpsinhλ(x:a) dμ−1(x)+sinhλ(1:a)∫Zpcoshλ(x:a) dμ−1(x), | (2.17) |
where a∈1λE∩E.
By solving the system of linear equations in (2.16) and (2.17), we obtain
∫Zpcoshλ(x:a) dμ−1(x)=1,∫Zpsinhλ(x:a) dμ−1(x)=−sinhλ(1:a)coshλ(1:a)+1, | (2.18) |
where a∈1λE∩E.
Thus, by (2.18) and Propositions 1 and 2, we get
∫Zpsinhλ(x:a) dμ−1(x)=−2sinhλ(12:a)coshλ(12:a)2cosh2λ(12:a)=−tanhλ(12:a), | (2.19) |
where a∈1λE∩E.
Therefore, by (2.18) and (2.19), we obtain the following theorem.
Theorem 6. Let a∈1λE∩E. Then we have the following relations.
∫Zpsinhλ(x:a) dμ−1(x)=−tanhλ(12:a),∫Zpcoshλ(x:a) dμ−1(x)=1. |
Thus, by Theorem 6, (2.13) and (1.12), we get
−tanhλ(12:a)=∫Zpsinhλ(x:a) dμ−1(x)=∞∑n=0(En,λ−(−1)nEn,−λ2)ann!,(a∈1λE∩E). | (2.20) |
In addition, by Theorem 6, (2.11) and (1.12), we get
1=∫Zpcoshλ(x;a) dμ−1(x)=∞∑n=0(En,λ+(−1)nEn,−λ2)ann!,(a∈1λE∩E). | (2.21) |
Thus, by(2.21), we get
12(En,λ+(−1)nEn,−λ)={1,ifn=0,0,ifn>0. | (2.22) |
For n≥1, we have
En,λ=(−1)n−1En,−λ. | (2.23) |
From (2.20) and (2.23), we obtain the following theorem.
Theorem 7. Let a∈1λE∩E. 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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