10,755 research outputs found

    Inequalities for generalized hypergeometric functions

    Get PDF
    AbstractIt is shown that some well-known Padé approximations for a particular form of the Gaussian hypergeometric function and two of its confluent forms give upper and lower bounds for these functions under suitable restrictions on the parameters and variable. With the aid of the beta and Laplace transforms, two-sided inequalities are derived for the generalized hypergeometric function pFq, p = q or p = q + 1, and for a particular form of Meijer's G-function. Several examples are developed. These include upper and lower bounds for certain elementary functions, complete elliptic integrals, the incomplete gamma function, modified Bessel functions, and parabolic cylinder functions

    NEW DOCTORAL DEGREES Integral expressions for series of functions of hypergeometric and Bessel types

    Get PDF
    This thesis presents some new results on integral expressions for series of functions of hypergeometric and Bessel types. Also there are derived two--sided inequalities of some hypergeometric functions, which are related with their integral representations. In the first part of the thesis are defined some special functions, mathematical methods, and results which we use in prooving our own. Some of them are Gamma function, Gauss hypergeometric function 2F1{}_2F_1 and generalized hypergeometric function pFq{}_pF_q. There are also Fox-Wright generalized hypergeometric function pPsiq{}_pPsi_q and the Struve function textbfHnu(z)textbf{H}_nu(z).\ Bessel differential equation is also described, and that is one of the crucial mathematical tools that we use.\ Mathieu (boldsymbola,boldsymbollambda)(boldsymbol a, boldsymbol lambda)-- and Dirichlet series are defined too, because they are useful for deriving most of integral representations. In that purpose, we also use condensed form of Euler--Maclaurin summation formula and fractional analysis, which are described in the introduction. In the middle part of the thesis, i.e. in Chapter 3, 4 and 5 we work on integral representations of functional series with members containing Bessel functions of the first kind, which are divided into three types: Neumann series, which are discussed in Chapter 3, Kapteyn series, which are described in Chapter 4, and Schl"omilch series, which are observed in Chapter 5. In the last chapter of this thesis, we obtain a functional series of hypergeometric types. There, we also derive an integral representations of hypergeometric functions, such as extended general Hurwitz--Lerch Zeta function and extended Hurwitz--Lerch Zeta function, and also the two-sided inequalities for the mentioned functions. \At the end of this chapter, new incomplete generalized Hurwitz--Lerch Zeta functions and incomplete generalized Gamma functions are defined, and we also investigate their important properties

    A class of completely monotonic functions involving divided differences of the psi and polygamma functions and some applications

    Full text link
    A class of functions involving the divided differences of the psi function and the polygamma functions and originating from Kershaw's double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving ratio of two gamma functions and originating from establishment of the best upper and lower bounds in Kershaw's double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in Rn1\mathbb{R}^{n-1} and Rn\mathbb{R}^n respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized.Comment: 11 page

    New Upper Bounds in the Second Kershaw's Double Inequality and its Generalizations

    Get PDF
    In the paper, new upper bounds in the second Kershaw’s double inequality and its generalizations involving the gamma, psi and polygamma functions are established, some known results are refined

    A Class of Logarithmically Completely Monotonic Functions and Application to the Best Bounds in the Second Gautschi-Kershaw's Inequality

    Get PDF

    On the expansion of the Kummer function in terms of incomplete Gamma functions

    Full text link
    The expansion of Kummer's hypergeometric function as a series of incomplete Gamma functions is discussed, for real values of the parameters and of the variable. The error performed approximating the Kummer function with a finite sum of Gammas is evaluated analytically. Bounds for it are derived, both pointwisely and uniformly in the variable; these characterize the convergence rate of the series, both pointwisely and in appropriate sup norms. The same analysis shows that finite sums of very few Gammas are sufficiently close to the Kummer function. The combination of these results with the known approximation methods for the incomplete Gammas allows to construct upper and lower approximants for the Kummer function using only exponentials, real powers and rational functions. Illustrative examples are provided.Comment: 21 pages, 6 figures. To appear in "Archives of Inequalities and Applications
    corecore