EIREducational Innovation Research3029-18443029-1852WHIOCE PUBLISHING PTE. LTD.10.18063/EIR.v4i3.1808ArticleIdentities Involving Hyperbolic Functions and Chebyshev Polynomialshttps://artdesignp.com/journal/EIR/4/3/10.18063/EIR.v4i3.1808DuShengsheng2026432026-03-26This paper discusses the power-sum problems of hyperbolic since and cosine functions by using the fundamental properties of Chebyshev polynomials of the first and second kinds. Based on their recurrence relations, explicit forms and related functional identities, we derive a set of combinatorial summation formulas for power sums of sinh(x) and cosh(x). The theorems and corollaries presented in this work further extend the known results concerning special polynomials and hyperbolic functions. These identities are applicable to number theory and combinatorial analysis, and enrich the practical applications of Chebyshev polynomials in summation problems.Hyperbolic functions,Chebyshev polynomials,Power index sums

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