Keyword: Generalized Gregory’s series Trigonometric identities Odd harmonic numbers In this paper the countable family of generalized Gregory’s series (all of which are the power series having the radius of convergence equal to one) is defined. Sums SnðxÞ; n 1⁄4 1;2; . . . of these series are found – first in the definite integral form and next in the form of finite linear combination of arcus ...

Abstract. In this paper we are concerned with a multiplier ω(n) of the Progressive means, and convex maps of the unit disc. With this concern we would have brought up in a rather unified approach the results of G. Pólya and I. J. Schoenberg in [7], T. Başgöze, J. L. Frank, and F. R. Keogh in [3], and Ziad S. Ali in [1]. More theorems on the properties of the multiplier ω(n) are given, and a key...

On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. In each case, there are three possible interpretations for the double series. In an earlier paper, two of the present authors proved the first identity under one possible interpretation. In the present paper, the second identity i...

For the Hahn and Krawtchouk polynomials orthogonal on the set {0, . . . , N} new identities for the sum of squares are derived which generalize the trigonometric identity for the Chebyshev polynomials of the first and second kind. These results are applied in order to obtain conditions (on the degree of the polynomials) such that the polynomials are bounded (on the interval [0, N ]) by their va...

In this paper we present some interesting results involving summation of series in particular trigonometric ones. We failed to locate these results in existing literature or in the web like MathWorld (http://mathworld.wolfram.com/) nor could we derive them using software for analytical computation like Maple. The identities are beautiful and involve finite series.

In contrast to Fourier series, these finite power sums are over the angles equally dividing the upper-half plane. Moreover, these beautiful and somewhat surprising sums often arise in analysis. In this note, we extend the above results to the power sums as shown in identities (17), (19), (25), (26), (32), (33), (34), (35), and (36) and in the appendix. The method is based on the generating func...

We prove a sharp bilinear estimate for the one dimensional Klein– Gordon equation. The proof involves an unlikely combination of five trigonometric identities. We also prove new estimates for the restriction of the Fourier transform to the hyperbola, where the pullback measure is not assumed to be compactly supported.