scholarly journals On a family of logarithmic and exponential integrals occurring in probability and reliability theory

1994 ◽  
Vol 35 (4) ◽  
pp. 469-478 ◽  
Author(s):  
M. Aslam Chaudhry
Keyword(s):  
Closed Form ◽  
Recurrence Relation ◽  
Reliability Theory ◽  
Integral Function ◽  
Integral Representations ◽  
Exponential Integral ◽  
Error Functions ◽  
Special Cases ◽  
Image Position ◽  
Form Evaluation

AbstractWe define an integral function Iμ(α, x; a, b) for non-negative integral values of μ byIt is proved that Iμ(α, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals. New integral representations of the exponential integral and complementary error functions are found as special cases.

On some infinite integrals involving logarithmic exponential and powers

1992 ◽  
Vol 122 (1-2) ◽  
pp. 11-15 ◽  
Author(s):  
M. Aslam Chaudhry ◽  
Munir Ahmad
Keyword(s):  
Recurrence Relation ◽  
Integral Function ◽  
Special Cases ◽  
Infinite Integral ◽  
Image Position

SynopsisIn this paper we define an integral function Iμ(α; a, b) for non-negative integral values of μ byIt is proved that the function Iμ(α; a, b) satisfies a functional recurrence-relation which is then exploited to evaluate the infinite integralSome special cases of the result are also discussed.

scholarly journals Direct Bijective Computation of the Generating Series for 2 and 3-Connection Coefficients of the Symmetric Group

10.37236/3226 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Alejandro H. Morales ◽  
Ekaterina A. Vassilieva
Keyword(s):  
Symmetric Group ◽  
Closed Form ◽  
Main Ingredient ◽  
Irreducible Characters ◽  
Long Cycle ◽  
Connection Coefficients ◽  
Special Cases ◽  
Generating Series ◽  
Orientable Surfaces ◽  
Form Evaluation

We evaluate combinatorially certain connection coefficients of the symmetric group that count the number of factorizations of a long cycle as a product of three permutations. Such factorizations admit an important topological interpretation in terms of unicellular constellations on orientable surfaces. Algebraic computation of these coefficients was first done by Jackson using irreducible characters of the symmetric group. However, bijective computations of these coefficients are so far limited to very special cases. Thanks to a new bijection that refines the work of Schaeffer and Vassilieva, we give an explicit closed form evaluation of the generating series for these coefficients. The main ingredient in the bijection is a modified oriented tricolored tree tractable to enumerate. Finally, reducing this bijection to factorizations of a long cycle into two permutations, we get the analogue formula for the corresponding generating series.

On the integration of incomplete elliptic integrals

1994 ◽  
Vol 444 (1922) ◽  
pp. 525-532 ◽  
Keyword(s):  
Closed Form ◽  
Hypergeometric Functions ◽  
Elliptic Integrals ◽  
Fractional Integrals ◽  
Special Cases ◽  
Complete Elliptic Integrals ◽  
Incomplete Elliptic Integrals ◽  
Form Evaluation

We give a closed-form evaluation of Erdélyi-Kober fractional integrals, involving incomplete elliptic integrals of the first kind, F ( φ, k ), and of the second kind, E ( φ, k ), which are integrated either with respect to the modulus or the amplitude. This is made possible by representing F ( φ, k ) and E ( φ, k ) in terms of the Kampé de Fériet double hypergeometric functions. Reduction formulae for these enable us to simplify the solutions for thirteen special cases, including integrals involving complete elliptic integrals. The hypergeometric character of the incomplete integrals is useful for evaluations of other classes of integrals involving F ( φ, k ) and E ( φ, k ).

scholarly journals A New Approach to the Analytic Evaluation of Thermonuclear Reaction Rates

1981 ◽  
Vol 93 ◽  
pp. 317-317
Author(s):  
H. J. Haubold ◽  
R. W. John
Keyword(s):  
Closed Form ◽  
Reaction Rate ◽  
Special Function ◽  
Reaction Rates ◽  
Thermonuclear Reaction ◽  
Fusion Plasma ◽  
New Approach ◽  
Analytic Evaluation ◽  
Image Position ◽  
Form Evaluation

In our paper (Haubold and John 1978) we succeeded in a closed-form evaluation of the reaction rate by means of a special function, known as Meijer's G-function This representation of the rate is appropriate to perform analytical operations (e.g. for the computation of energy generation in a fusion plasma).

scholarly journals An Interpolation Series for Integral Functions

1953 ◽  
Vol 9 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Sheila Scott Macintyre
Keyword(s):  
Taylor Expansion ◽  
Maximum Modulus ◽  
Integral Function ◽  
The Best Possible Constant ◽  
Special Cases ◽  
Interpolation Series ◽  
Image Position

1. The Gontcharoff interpolation serieswherehas been studied in various special cases. For example, if an = a0 (all n), (1.0) reduces to the Taylor expansion of F(z). If an = (−1)n, J. M. Whittaker showed that the series (1.0) converges to F(z) provided F(z) is an integral function whose maximum modulus satisfiesthe constant ¼π being the “best possible”. In the case |an| ≤ 1, I have shown that the series converges to F(z) provided F(z) is an integral function whose maximum modulus satisfiesand that while ·7259 is not the “best possible” constant here, it cannot be replaced by a number as great as ·7378.

scholarly journals Some triple integral equations

1960 ◽  
Vol 4 (4) ◽  
pp. 200-203 ◽  
Author(s):  
C. J. Tranter
Keyword(s):  
Boundary Condition ◽  
Closed Form ◽  
Integral Equations ◽  
Dual Integral Equations ◽  
Triple Integral Equations ◽  
Special Cases ◽  
Potential Problems ◽  
Image Position ◽  
Different Parts

Potential problems in which different conditions hold over two different parts of the same boundary can often be conveniently reduced to the solution of a pair of dual integral equations. In some problems, however, the boundary condition is such that different conditions hold over three different parts of the boundary and, in such cases, the integral equations involved are frequently of the formwhere f(r), g(r) are specified functions of r, p = ± ½ and ø(u) is to be found. Such equations might well be called triple integral equations and, in this note, I point out certain special cases which I have found to be capable of solution in closed form.

scholarly journals The incomplete Srivastava’s triple hypergeometric functions γHB and ΓHB

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1779-1787 ◽  
Author(s):  
Junesang Choi ◽  
Rakesh Parmar ◽  
Purnima Chopra
Keyword(s):  
Recurrence Relation ◽  
Hypergeometric Functions ◽  
Integral Representations ◽  
Reduction Formula ◽  
Gamma Functions ◽  
Fundamental Properties ◽  
Special Cases ◽  
Incomplete Gamma Functions ◽  
Derivative Formula

Recently Srivastava et al. [26] introduced the incomplete Pochhammer symbols by means of the incomplete gamma functions ?(s,x) and ?(s,x), and defined incomplete hypergeometric functions whose a number of interesting and fundamental properties and characteristics have been investigated. Further, ?etinkaya [6] introduced the incomplete second Appell hypergeometric functions and studied many interesting and fundamental properties and characteristics. In this paper, motivated by the abovementioned works, we introduce two incomplete Srivastava?s triple hypergeometric functions ?HB and ?HB by using the incomplete Pochhammer symbols and investigate certain properties, for example, their various integral representations, derivative formula, reduction formula and recurrence relation. Various (known or new) special cases and consequences of the results presented here are also considered.

XVIII.—The Definite Integrals of Interpolation Theory

1931 ◽  
Vol 50 ◽  
pp. 220-224
Author(s):  
E. T. Copson
Keyword(s):  
Infinite Series ◽  
Interpolation Formula ◽  
Integral Function ◽  
Integral Representations ◽  
Definite Integral ◽  
Finite Region ◽  
Interpolation Theory ◽  
Definite Integrals ◽  
Uniformly Convergent ◽  
Image Position

It is well known that, if the infinite seriesis convergent for any non-integral value of z, it is uniformly convergent in any finite region of the z-plane and represents an integral function C(z), say, such that C(n) = an for n = 0, ± 1, ± 2, … It is called the Cardinal Function of the table of values, and is identical with Gauss's Interpolation Formula (suitably bracketed).The function C(z) defined by the series (1) has been given two different definite integral representations, due to Ferrar and Ogura respectively.

scholarly journals A remark on “Uniform O– estimates of certain error functions connected with k-free integers”

1973 ◽  
Vol 15 (2) ◽  
pp. 177-178 ◽  
Author(s):  
D. Suryanarayana
Keyword(s):  
Error Functions ◽  
Image Position

In this note I follow the same notation adopted in [2]. Let Qk(x, n) denote the number of k-free integers ≦x which are prime to n. In [2], I proved the following: For o≦s>1/k uniformly, where σ*−s(n) is the sum of the reciprocals of the sth powers of the square-free divisor of n.

Closed form evaluation of sums containing squares of Fibonomial coefficients

2016 ◽  
Vol 66 (3) ◽  
Author(s):  
Emrah Kiliç ◽  
Helmut Prodinger
Keyword(s):  
Closed Form ◽  
Systematic Approach ◽  
Sums Of Squares ◽  
Lucas Numbers ◽  
Fibonomial Coefficients ◽  
Finite Products ◽  
Fibonacci And Lucas Numbers ◽  
Form Evaluation

AbstractWe give a systematic approach to compute certain sums of squares of Fibonomial coefficients with finite products of generalized Fibonacci and Lucas numbers as coefficients. The technique is to rewrite everything in terms of a variable

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