Integral representations of Euler's type for certain generalized Horn functions

Several integral representations of Euler's type for the generalized Horn functions ^(k)H₃⁽ⁿ⁾ and ^(k)H₄⁽ⁿ⁾ are given; and quadratic transformations expressing these as F_D⁽ⁿ⁾ and F_A⁽ⁿ⁾, respectively, are derived. Moreover, a reduction formula expressing ^(k)H₃⁽ⁿ⁾ in terms of ⁽¹⁾H₃⁽ⁿ⁾ is given.     

Norm inequalities for a certain class of ∞ functions

In 1936 the author showed that the function sin(π(x+1)/4) is the entire function of least exponential type (=π/4) among all entire functionsf(z) with the property thatf ⁽ⁿ⁾(z) vanishes somewhere in the real interval [−1, 1] (n=0, 1,2,…). Now more precise results of this kind are obtained by working within the class ∞[−1, 1]. For Paul Montel on his 95th birthday     

On a certain class of modular functions

We give a characterization of those meromorphic modular functions on a subgroup of finite index of the full modular group whose divisors are supported at the cusps, in terms of the growth of the exponents of their infinite product expansions.

     

Starlikeness of certain functions with integral representations

This paper is concerned with the starlikeness of certain normalized functions, analytic in the open unit disk, defined by integral representations. Results due to Caplinger and Singh are extended.     

Series representations for special functions and mathematical constants

We develop a collection of single-parameter series representations involving special functions and mathematical constants. The techniques used result in new representations as well as alternative proofs of known representations.     

The Weyl functions of a certain class of Sturm-Liouville operators

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 52, pp. 59–69, 1989.     

On a special class of convex functions

We answer in the affirmative to a conjecture concerning convex functions.     

ON INTERPOLATION OF A CERTAIN CLASS OF FUNCTIONS

I.StatementofResultsForarealaleta~ma-c(1,Ial).Lets21.WeuseG.todenotethes-dimensionalunitcube,and(:,')todenotethescalarproductofvectors:and,inH.Form(ml,',m.)EZsand^~(Al,',A.)EHwedenotel'II^=mtl...m:a.Inparticular,ifA=(A,',A),thenwewritellmJI'=IImll^.Fulthermore,forf=(TI,'.)r.)EH,letiff=lrlI ... Ir.].Letf~(n,',r.)ERswithfi20(i~1,',s),andletf(:)beasingle--valuedfunctionsuchthatf(xl,'sxit',x.)~f(xl,')xi 1,',x.),i~1,')s.Writefi=pi fi(i~1,',s),wherepi~fi~0iffi=0an…     

Unitarizability of a certain class of irreducible representations of classical groups

We prove that a certain class of irreducible representations of the classical p-adic groups is unitarizable and in general, can be isolated in the unitary dual. These representations are Aubert duals of a certain class of square-integrable representations, thus, in this case, Bernstein’s conjecture, which states that the Aubert involution preserves unitarizability, is confirmed.     

On chebyshev quadrature for a special class of weight functions

Recently Gautschi [5] has proved that the weight functionsw _ξ defined by $$w_\xi (x) = \left\{ {\begin{array}{*{20}c} {\left| x \right|(x^2 - \xi ^2 )^{ - 1/2} (1 - x^2 )^{ - 1/2} ,x \in [ - 1, - \xi ] \cup [\xi ,1]} \\ {0\,\,elsewhere,\,0< \xi< 1} \\ \end{array} } \right.$$ are the only symmetric ones, apart from the Chebyshev weight function, for which there exists for every evenn a Chebyshev rule in the strict sense, havingn nodes and Gaussian degree 2n?1. In this note we show that for an odd numbern of nodes the maximum degree of Chebyshev-type rules forw _ξ has a completely different behavior from that for evenn.     

Closed form representations for a class of compactly supported radial basis functions

In this paper we re-examine Wendland’s strategy for the construction of compactly supported positive definite radial basis functions. We acknowledge that this strategy can be modified to capture a much larger range of functions, including the so-called missing Wendland functions which have been the subject of a recent paper by Schaback (Adv Comput Math 34:67–81, 2011). Our approach is to focus on a general integral representation of such functions and we will show how a careful evaluation of this integral leads to new closed form expressions for both Wendland’s original functions and the missing ones. The resulting expressions are easy to code and so provide the potential user with a quick way of accessing a desired example for a given application.     

Asymptotic expansions for the zeros of certain special functions

We derive asymptotic expansions for the zeros of the cosine-integral Ci(x) and the Struve function $\text{H}{}_0\text{(x)}$, and extend the available formulae for the zeros of Kelvin functions. Numerical evidence is provided to illustrate the accuracy of the expansions.