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In the paper a number of identities involving even powers of the values of functions tangent, cotangent, secans and cosecans are proved. Namely, the following relations are shown: [wzory] where m, n are positive integers, f is one of the functions: tangent, cotangent, secans or cosecans and wf(x),vf(x),~wf(x) are some polynomials from Q[x]. One of the remarkable identities is the following: [wzór] Some of these identities are used to find, by elementary means, the sums of the series of the form [wzór] , where n is a fixed positive integer. One can also notice that Bernoulli numbers appear in the leading coeficients of the polynomials wf(x),vf(x) and ~ wf(x).
Czasopismo
Rocznik
Tom
Strony
39--50
Opis fizyczny
Bibliogr. 8 poz.
Twórcy
autor
- Institute of Mathematics, Pedagogical University, ul. Podchorążych 2, PL-30-084 Kraków, Poland
autor
- Pawel Wlodkowic University College in Plock, al. Kilińskiego 12, PL-09-402 Plock, Poland
Bibliografia
- [1] T. M. Apostol, Another elementary proof of Euler's formula for ζ(2n), Amer. Math. Monthly 80 (1973), No. 4, 425-431.
- [2] P.M. Cohn, Algebra. Vol. 1, Second edition, John Wiley & Sons, Ltd., Chichester, 1982.
- [3] D. Cvijovic, J. Klinowski, Finite cotangent sums and the Riemann zeta function, Math. Slovaca 50 (2000), no. 2, 149-157.
- [4] R.L. Graham, D.E. Knuth, O. Patashnik, Concrete mathematics. A foundation for computer science, Second edition, Addison-Wesley Publishing Company, Reading, MA, 1994.
- [5] R.A. Kortram, Simple proofs for... Math. Mag. 69 (1996), no. 2, 122-125.
- [6] I. Papadimitriou, A simple proof of the formula [wzór], Amer. Math. Monthly 80 (1973), No. 4, 424-425.
- [7] T.J. Ransford, An elementary proof of [wzór], Eureka No. 42, Summer 1982, 3-5.
- [8] H. Tsumura, An elementary proof of Euler's formula for ζ(2m), Amer. Math. Monthly 111 (2004), no. 5, 430-431.
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Bibliografia
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