Genetic programming (GP) may be seen as a machine learning method, which induces a population of computer programs by evolutionary means (Banzhaf et al. 1998). Genetic programming has been used successfully in generating computer programs for solving a number of problems in a wide range of areas. In (Hoai and McKay 2001), we proposed a framework for a grammar-guided genetic programming system c...

Abstract Legendre was the first to evaluate two well-known integrals involving sines and exponentials. One of these can be used prove Binet’s second formula for logarithm gamma function. Here, we show that other integral leads a specific case Hermite’s generalization formula. From analogs Legendre’s integrals, with replaced by cosines, obtain integration identities logarithms trigonometric func...

In this paper, using the number of spanning trees in some classes of graphs, we prove the identities: Fn = 2n−1 n √

In this paper we introduce an algebraic integer encoding scheme for the basis matrix elements of DCTs and IDCTs. In particular, we encode the function and generate the other matrix elements using standard trigonometric identities. This encoding technique eliminates the requirement to approximate the matrix elements; rather we use algebraic ‘placeholders’ for them. Using this encoding scheme we ...

1. R. L. Adler & T. J. Rivlin. "Ergodic and Mixing Properties of Chebyshev Polynomials." Proa. Amer. Math. Soc. 15 (1964) :79'4-7'96. 2. P. Johnson & A. Sklar. "Recurrence and Dispersion under Iteration of Cebysev Polynomials." To appear. 3. C.H. Kimberling. "Four Composition Identities for Chebyshev Polynomials." This issue, pp. 353-369. 4. T. J. Rivlin. The Chebyshev Polynomials. New York: Wi...

The Chebyshev polynomials have many beautiful properties and countless applications, arising in a variety of continuous settings. They are a sequence of orthogonal polynomials appearing in approximation theory, numerical integration, and differential equations. In this paper we approach them instead as discrete objects, counting the sum of weighted tilings. Using this combinatorial approach, on...

Inspired by Rearick (1968), we introduce two new operators, LOG and EXP. The LOG operates on generalized Fibonacci polynomials giving generalized Lucas polynomials. The EXP is the inverse of LOG. In particular, LOG takes a convolution product of generalized Fibonacci polynomials to a sum of generalized Lucas polynomials and EXP takes the sum to the convolution product. We use this structure to ...

Motivated by the substantial development of special functions, we contribute to establish some rigorous results on general series identities with bounded sequences and hypergeometric functions different arguments, which are generally applicable in nature. For application purpose, apply our e.g. Trigonometric Elliptic integrals, Dilogarithmic function, Error Incomplete gamma many other functions.

This article concerns the examination of trigonometric identities from an elliptic perspective. The treatment functions presented herein adheres to a structure analogous traditional exposition functions, with exception that ellipse replaces unit circle. degree similarity between and their counterparts is moderated by periodicity so-called El- functions. These not only establish values but also ...