Evaluations of some classes of the trigonometric moment integrals

Four classes of the trigonometric moment integrals are evaluated in closed form in a simple and unified manner by making use of the contour integration in conjunction with the Cauchy integral theorem. In all cases, the closed contour of the same shape is used and it is shown that the integrals are expressible only in terms of the Hurwitz zeta function and elementary functions. A number of interesting (known or new) special cases and consequences of the main results are also considered.     

Some reformulated properties of Jacobian elliptic functions

Various properties of Jacobian elliptic functions can be put in a form that remains valid under permutation of the first three of the letters c, d, n, and s that are used in pairs to name the functions. In most cases 12 formulas are thereby replaced by three: one for the three names that end in s, one for the three that begin with s, and one for the six that do not involve s. The properties thus unified in the present paper are linear relations between squared functions (16 relations being replaced by five), differential equations, and indefinite integrals of odd powers of a single function. In the last case the unification entails the elementary function RC(x,y)=RF(x,y,y), where RF(x,y,z) is the symmetric elliptic integral of the first kind. Explicit expressions in terms of RC are given for integrals of first and third powers, and alternative expressions are given with RC replaced by inverse circular, inverse hyperbolic, or logarithmic functions. Three recurrence relations for integrals of odd powers hold also for integrals of even powers.     

Symmetry in c, d, n of Jacobian elliptic functions

The relation connecting the symmetric elliptic integral RF with the Jacobian elliptic functions is symmetric in the first three of the four letters c, d, n, and s that are used in ordered pairs to name the 12 functions. A symbol Δ(p,q)=ps²(u,k)−qs²(u,k), p,q∈{c,d,n}, is independent of u and allows formulas for differentiation, bisection, duplication, and addition to remain valid when c, d, and n are permuted. The five transformations of first order, which change the argument and modulus of the functions, take a unified form in which they correspond to the five nontrivial permutations of c, d, and n. There are 18 transformations of second order (including Landen's and Gauss's transformations) comprising three sets of six. The sets are related by permutations of the original functions cs, ds, and ns, and there are only three sets because each set is symmetric in two of these. The six second-order transformations in each set are related by first-order transformations of the transformed functions, and all 18 take a unified form. All results are derived from properties of RF without invoking Weierstrass functions or theta functions.     

On some definite integrals and infinite sums containing products of special functions

A method is proposed for evaluation of some definite integrals and infinite sums containing products of Bessel, Struve and other special functions.     

Numerical evaluation of finite Fourier integrals

A computationally efficient algorithm for evaluating Fourier integrals ∫¹_?1?(x)e^iωxdx using interpolatory quadrature formulas on any set of collocation points is presented. Examples are given to illustrate the performances of interpolatory formulas which are based on the applications of the Fejér, Clenshaw—Curtis, Basu and the Newton—Cotes points. Initially, the formulas for nonoscillatory integrals are generated and then generalizations to finite Fourier integrals are made. Extensions of this algorithm to some other weighted integrals are also considered.     

On multiple integrals of special form

Multiple integrals generalizing the iterated kernels of integral operators are expressed as single integrals in the case of a special representation of the kernel (this is our theorem). Besides integral equations, Markov processes involve these integrals as well. As a consequence of the theorem, we obtain transition probability densities of certain Markov processes. As an illustration, we consider nine examples.     

Generalization of some hypergeometric functions

In this paper, we make use of the Lagrange interpolation to obtain some algebraic identities, involving one or two infinite sets of variables.     

On the relationship between the Kolmogorov and Henstock integrals

The Henstock integrals are proved to be narrower than the Kolmogorov integral. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 832–844, December, 1996. In conclusion, it should be mentioned that this work is stimulated by the author's personal conversation with Proffessor V. Skvortsov, who pointed out the significance of studying the relationship between the Kolmogorov and Henstok integrals.     

A Cauchy-Schwarz type inequality for fuzzy integrals

In this paper we prove a Cauchy-Schwarz type inequality for fuzzy integrals.     

A Chebyshev type inequality for fuzzy integrals

In this paper, we prove a Chebyshev type inequality for fuzzy integrals. More precisely, we show that:

where μ is the Lebesgue measure on and f,g:[0,1]→[0,∞) are two continuous and strictly monotone functions, both increasing or both decreasing. Also, some examples and applications are presented.     

A note on moment integrals and some applications

In this paper, two moment integrals are given by the method of transformation. In addition, generalizations of Sears?s transformation are obtained by moment integrals. Moreover, certain q  -Mehler formulas for Rogers–Szegö polynomials are gained by moment integrals. Besides, an open problem of trilinear generating function is deduced by moment integrals. At last, generalizations of U(n+1)$U(n+1)$ type Kalnins–Miller transformation are achieved by moment integrals.     

On the two-parameter fractional Brownian motion and Stieltjes integrals for Hölder functions

We extend the Stieltjes integral to Hölder functions of two variables and prove an existence and uniqueness result for the corresponding deterministic ordinary differential equations and also for stochastic equations driven by a two-parameter fractional Brownian motion.     

Content evaluation and class symmetric functions

In this article we study the evaluation of symmetric functions on the alphabet of contents of a partition. Applying this notion of content evaluation to the computation of central characters of the symmetric group, we are led to the definition of a new basis of the algebra Λ of symmetric functions over that we call the basis of class symmetric functions.By definition this basis provides an algebra isomorphism between Λ and the Farahat-Higman algebra FH governing for all n the products of conjugacy classes in the center of the group algebra of the symmetric group . We thus obtain a calculus of all connexion coefficients of inside Λ. As expected, taking the homogeneous components of maximal degree in class symmetric functions, we recover the symmetric functions introduced by Macdonald to describe top connexion coefficients.We also discuss the relation of class symmetric functions to the asymptotic of central characters and of the enumeration of standard skew young tableaux. Finally we sketch the extension of these results to Hecke algebras.     

Algebraic independence of logarithmic singularities of some complex functions

We show that algebraic independence of some complex functions of one variable over regular functions implies their algebraic independence over a larger ring, containing complex powers of regular functions. Based on this we obtain a generalization of a special case of the theorem of Kaczorowski and Perelli on functional independence of logarithms of functions in the Selberg class. As an application we state a new result on oscillations of arithmetical functions.     

Convergence in mean of some random Fourier series

For a symmetric stable process X(t,ω) with index α∈(1,2], f∈Lp[0,2π], p?α, and , we establish that the random Fourier-Stieltjes (RFS) series converges in the mean to the stochastic integral , where fβ is the fractional integral of order β of the function f for . Further it is proved that the RFS series is Abel summable to . Also we define fractional derivative of the sum of order β for an, An(ω) as above and . We have shown that the formal fractional derivative of the series of order β exists in the sense of mean.     

Asymmetric integral as a limit of generated Choquet integrals based on absolutely monotone real set functions

We introduce a generated Choquet integral with respect to absolutely monotone and sign stable set functions. Another type of integrals defined with respect to such a set function is obtained as a limit of an appropriate sequence of generated Choquet integrals.     

Equivalence of some integrals and modulus of smoothness

Let f∈C(R). We are interested in lower and upper bounds of the integrals

     

On an extremal problem of Selberg

We consider the minimum value m_p of the product under the conditions that and . This problem stems from estimation of the Selberg integral appeared in the study of integral-valued entire functions [A. Selberg, Über einen Satz von A. Gelfond, Arch. Math. Naturvid. 44 (1941) 159–170]. We give upper and lower estimates of m_p : . This, in particular, slightly improves the earlier lower estimate given by Selberg: logm_p>p·0.31654924….     

On certain generalized families of unified elliptic-type integrals pertaining to Euler integrals and generating functions

Elliptic-type integrals have their importance and potential in certain problems in radiation physics and nuclear technology. A number of earlier works on the subject contains several interesting unifications and generalizations of some significant families of elliptic-type integrals. The present paper is intended to obtain certain new theorems on generating functions. The results obtained in this paper are of manifold generality and basic in nature. Beside deriving various known and new elliptic-type integrals and their generalizations these theorems can be used to evaluate various Euler-type integrals involving a number of generating functions.