Parametric integrals, the Catalan numbers, and the beta function
Summary: In mathematics, computing definite integrals and improper integrals is important. The beta and gamma functions, also called the Euler integrals of the first and second kinds, can be defined by definite (improper) integrals and are most extensively-applied special functions. The Catalan numbers form a sequence of natural numbers that occur in various counting problems in combinatorial mathematics. The Wallis ratio is connected with the ratio of two gamma functions, with the ratio of two double factorials, and with definite integral whose integrand is the power of sine or cosine and has applications in statistics. In the paper, starting from a sequence of parametric integrals, the author presents a unified expression of two known results on the sequence of parametric integrals in terms of the gamma function, computes a new sequence of parametric integrals in terms of the gamma function, finds the absolute convexity of the Catalan numbers, calculates a general sequence of parametric integrals in terms of the beta function, and represents the above parametric integrals, the Catalan numbers, and the Wallis ratio in terms of the beta function.
scientific article; zbMATH DE number 6808302