Remarkable identities

• Autorzy: Górowski J. , Żabowski J.

Identyfikatory

DOI

10.7862/rf.2014.4

• Języki publikacji: EN

Abstrakty

EN

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).

Słowa kluczowe

EN

trigonometric identity; zeta function

PL

tożsamość trygonometryczna; funkcja zeta

• Wydawca: Oficyna Wydawnicza Politechniki Rzeszowskiej
• Czasopismo: Journal of Mathematics and Applications
• Rocznik: 2014
• Tom: Vol. 37
• Strony: 39--50
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Górowski J.

• Institute of Mathematics, Pedagogical University, ul. Podchorążych 2, PL-30-084 Kraków, Poland

autor

Żabowski J.

• Pawel Wlodkowic University College in Plock, al. Kilińskiego 12, PL-09-402 Plock, Poland

Bibliografia

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• [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.