We list the so-called Askey-scheme of hypergeometric orthogonal polynomials and we give a q- analogue of this scheme containing basic hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation, the second order di#erential or di#erence equation, the forward and backward shift operator, the Rodrigues-type formula and generating functions of all classes of orthogonal polynomials in this scheme. In chapter 2 we give the limit relations between di#erent classes of orthogonal polynomials listed in the Askey-scheme. In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation, second order di#erence equation, forward and backward shift operator, Rodrigues-type formula and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally, in chapter 5 we...

We present new inequalities for the gamma and psi functions, and we provide new classes of completely monotonic, star-shaped, and superadditive functions which are related to Γ and ψ. Euler’s gamma function Γ(x) =

We obtain such upper bounds for Jacobi polynomials which are uniform in all the parameters involved and which contain explicit constants. This is done by a combination of some results on generalized Christo#el functions and some estimates of Jacobi polynomials in terms of Christo#el functions. 1.

The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.

Abstract. In the survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some known results, including Wendel’s, Gurland’s, Kazarinoff’s, Gautschi’s, Watson’s, Chu’s, Lazarević-Lupa¸s’s, Kershaw’s and Elezović-Giordano-Pečarić’s inequalities, claim, monotonic and convex properties. On the other hand, we introduce some related advances on the topic of bounding the ratio of two gamma functions in recent years. Contents

k (x) = O max 1, (α 2 + β 2) 1/4}) max x∈[−1,1] (1 − x)α+ 1 2 (1 + x) β+1 2 [Erdélyi et al.,Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602-614]. Here we will confirm this conjecture in the ultraspherical case α = β ≥ 1+√2, even in a stronger 4 form by giving very explicit upper bounds. We also show that δ2 − x2 2 α (1 − x) ( P (α,α)