7 Periodic Series and Functions 128 7.1 Basic Trigonometric Identities . . . . . . . . . . . . . . . . . . 128 7.2 Basic Trigonometric Integrals . . . . . . . . . . . . . . . . . . 130 7.3 Basic Properties of Complex Numbers . . . . . . . . . . . . . 131 7.4 Complex Numbers and Trigonometrical Identities . . . . . . . 133 7.5 The Exponential of a Complex Number . . . . . . . . . . . . . 133 7.6...

Webb & Parberry proved in 1969 a startling trigonometric identity involving Fibonacci numbers. This identity has remained isolated up to now, despite the amount of work on related polynomials. We provide a wide generalization of this identity together with what we believe (and hope!) to be its proper understanding.

1 Trigonometric Functions We recall the following identities for trigonometric functions. Theorem 1.1. For all x ∈ R, cos(−x) = cos(x) and sin(−x) = − sin(x). Theorem 1.2. For all α, β ∈ R, cos(α + β) = cos(α) cos(β)− sin(α) sin(β). Theorem 1.3. For all α, β ∈ R, sin(α+ β) = sin(α) cos(β) + cos(α) sin(β). Proof. For small positive angles (so α+ β is less than a right angle), the identities from...

In this paper we will use one well-known modular equation of seventh order, one theta function identity of S. McCullough and L.-C. Shen, 1994, and the complex variable theory of elliptic functions to prove some new septic identities for theta functions. Then we use these identities to provide new proofs of some Eisenstein series identities in Ramanujan’s notebooks or “lost” notebook. We also de...

Abstract. One fragment (page 335) published with Ramanujan’s lost notebook contains two formulas, each involving a finite trigonometric sum and a doubly infinite series of Bessel functions. The identities are connected with the classical circle and divisor problems, respectively. This paper is devoted to the first identity. First, we obtain a generalization in the setting of Riesz sums. Second,...

The article describes some interesting and possibly not previously noticed nonlinear differential identities satisfied by exponential and trigonometric functions. Both formulation and the proofs of these relations do not require mathematics knowledge beyond standard differentiation rules and high school algebra. Surprisingly, the research that produced these identities was devoted to medical im...

We prove several old and new theorems about finite sums involving characters and trigonometric functions. These sums can be traced back to theta function identities from Ramanujan’s notebooks and were systematically first studied by Berndt and Zaharescu; their proofs involved complex contour integration. We show how to prove most of Berndt–Zaharescu’s and some new identities by elementary metho...

Based on Spiridonov's analysis of elliptic generalizations of the Gauss hypergeomet-ric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E 7 (ellip-tic, hyperbolic) and of type E 6...