In this article we derive some entries and errata for the book of Gradshteyn Ryzhik which were originally published by Bierens de Haan. We summarize our results using tables easy reading referencing.

In this paper, we first introduce the incomplete extended Gamma and Beta functions with matrix parameters; then, establish some different properties for these new extensions. Furthermore, give a specific application Bessel function by using functions; at last, construct relation between confluent hypergeometric function.

1. E. P. Merkes & W. T. Scott, "Continued fraction solutions of the Riccati equation," J. Math. Anal. Appl., v. 4, 1962, pp. 309-327. MR 25 #4167. 2. W. Fair, "Padé Approximations to the Solution of the Riccati Equation," Math. Comp., v. 18, 1964, pp. 627-634. MR 29 #6630. 3. Y. L. Luke, "The Padé table and the r-method," J. Math, and Phys., v. 37, 1958, pp. 110-127. MR 20 #5558. 4. H. T. Davis...

I consider an arbitrary-precision computation of the incomplete Gamma function from the Legendre continued fraction. Using the method of generating functions, I compute the convergence rate of the continued fraction and find a direct estimate of the necessary number of terms. This allows to compare the performance of the continued fraction and of the power series methods. As an application, I s...

Let p 6= 1 be a positive real number. We determine all real numbers α = α(p) and β = β(p) such that the inequalities [1− e−βx p ] < 1 Γ(1 + 1/p) ∫ x 0 e−t p dt < [1− e−αx p ] are valid for all x > 0. And, we determine all real numbers a and b such that − log(1− e−ax) ≤ ∫ ∞ x e−t t dt ≤ − log(1− e−bx)

The incomplete Gamma function γ(α, x) and its associated functions γ(α, x+) and γ(α, x−) are defined as locally summable functions on the real line and some convolutions and neutrix convolutions of these functions and the functions x and x − are then found.

Abstract. Using a variational approach, two new series representations for the incomplete Gamma function are derived: the first is an asymptotic series, which contains and improves over the standard asymptotic expansion; the second is a uniformly convergent series, completely analytical, which can be used to obtain arbitrarily accurate estimates of Γ(a, x) for any value of a or x. Applications ...

The objective of this paper is to define and study new generalized extended gamma functions. A generalized extended gamma probability density function involving generalized hypergeometric function is also defined. Closed form representations of the generalized gamma functions and the moment generating function are derived in the form of H-function using inverse Mellon transform techniques. Inco...