By using trigonometric and generating functions, some formulas and relations involving sums of powers of consecutive positive integers and certain combinatorial sums are derived. By applying the derivative operator to some certain families of special functions and finite sums involving trigonometric functions, many novel relations related to the special numbers and polynomials are obtained. Moreover, by applying p-adic integrals to these finite sums, some p-adic integral representations of trigonometric functions are found. [ABSTRACT FROM AUTHOR]

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