Some identities on degenerate hyperbolic functions arising from $ p $-adic integrals on $ mathbb{Z}_p $

Some identities on degenerate hyperbolic functions arising from $ p $-adic integrals on $ mathbb{Z}_p $

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The aim of this paper is to introduce several degenerate hyperbolic functions as read more 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 here degenerate Euler numbers and the Cauchy numbers of the first kind.