scholarly journals The Properties of a new Orthogonal Function Associated with the Confluent Hypergeometric Function

1924 ◽  
Vol 43 ◽  
pp. 117-130
Author(s):  
George E. Chappell
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Hypergeometric Function ◽  
Special Form ◽  
Confluent Hypergeometric Function ◽  
Orthogonal Function ◽  
Image Position

The function whose properties are discussed in this note, is a special form of Whittaker's Confluent Hypergeometric Function, Wkm(z). It is the general solution of the Differential Equationand can be obtained in the form of a series, terminating only when k is half a positive odd integer, viz.,

A differential equation for undamped forced non-linear oscillations. III

1965 ◽  
Vol 61 (1) ◽  
pp. 133-155 ◽  
Author(s):  
G. R. Morris
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Periodic Solutions ◽  
Special Interest ◽  
Bounded Solutions ◽  
General Differential Equation ◽  
Dynamical Description ◽  
Image Position ◽  
Essential Problem ◽  
Special Case

The most general differential equation to which the dynamical description of the title applies iswhere dots denote differentiation with respect to t. The essential problem for this equation is to determine the behaviour of solutions as t → ∞. When we attack this problem, the most obvious question is whether, under reasonable conditions on p(t), every solution is bounded as t → ∞ this question is open except when g(x) is linear. In the special case when p(t) is periodic, (1·1) may have periodic solutions; it is clear that any such solution is bounded, and it is worth mentioning that finding periodic solutions is the easiest way of finding particular bounded ones. So long as the bounded-ness problem is unsolved, there is a special interest in finding a large class of particular bounded solutions: if we know such a class then, although we cannot say whether the general solution is bounded or not, we can make the imprecise comment that either the general solution is in fact bounded or the structure of the whole set of solutions is quite complicated.

scholarly journals The Elliptic Cylinder Functions of the Second Kind

1914 ◽  
Vol 33 ◽  
pp. 2-13 ◽  
Author(s):  
E. Lindsay Ince
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Differential Equations ◽  
Elliptic Cylinder ◽  
Linear Differential Equations ◽  
Periodic Functions ◽  
Linear Differential ◽  
Image Position

The differential equation of Mathieu, or the equation of the elliptic cylinder functionsis known by the theory of linear differential equations to have a general solution of the typeφ and ψ being periodic functions of z, with period 2π.

scholarly journals On the Relation between Pincherle's Polynomials and the Hypergeometric Function

1920 ◽  
Vol 39 ◽  
pp. 58-62 ◽  
Author(s):  
Bevan B. Baker
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Hypergeometric Function ◽  
The Difference ◽  
Image Position

1. The Pincherle polynomials are defined as the coefficients in the expansion of {1 − 3 tx + t3}−½ in ascending powers of t. If the coefficient of tn be denoted by Pn(x), then the polynomials satisfy the difference equationand Pn(x) satisfies the differential equation

The real zeros of the confluent hypergeometric function

1956 ◽  
Vol 52 (4) ◽  
pp. 626-635 ◽  
Author(s):  
L. J. Slater
Keyword(s):  
Hypergeometric Function ◽  
Numerical Evaluation ◽  
Confluent Hypergeometric Function ◽  
The Real ◽  
Real Zeros ◽  
Image Position

This paper contains a discussion of various points which arise in the numerical evaluation of the small real zeros of the confluent hypergeometric functionwhereThere are two distinct problems, first the determination of those values of x for which M(a, b; x) = 0, given a and b, and secondly the study of the curves represented by M (a, b; x) = 0, for fixed values of x. These curves all lie on the surface M(a, b; x) = 0, of course.

Estimates on the Green's function of second-order elliptic operators in ℝN

1998 ◽  
Vol 128 (5) ◽  
pp. 1033-1051
Author(s):  
Adrian T. Hill
Keyword(s):  
Green’S Function ◽  
Heat Kernel ◽  
Hypergeometric Function ◽  
Green's Function ◽  
Confluent Hypergeometric Function ◽  
Elliptic Operators ◽  
Polar Coordinates ◽  
Pointwise Estimates ◽  
Image Position ◽  
Second Order Elliptic Operators

Sharp upper and lower pointwise bounds are obtained for the Green's function of the equationfor λ> 0. Initially, in a Cartesian frame, it is assumed that . Pointwise estimates for the heat kernel of ut + Lu = 0, recently obtained under this assumption, are Laplace-transformed to yield corresponding elliptic results. In a second approach, the coordinate-free constraint is imposed. Within this class of operators, the equations defining the maximal and minimal Green's functions are found to be simple ODEs when written in polar coordinates, and these are soluble in terms of the singular Kummer confluent hypergeometric function. For both approaches, bounds on are derived as a consequence.

scholarly journals On the Volterra-Type Fractional Integro-Differential Equations Pertaining to Special Functions

2020 ◽  
Vol 4 (3) ◽  
pp. 33
Author(s):  
Yudhveer Singh ◽  
Vinod Gill ◽  
Jagdev Singh ◽  
Devendra Kumar ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Hypergeometric Function ◽  
Fractional Order ◽  
Integral Transform ◽  
Special Functions ◽  
Confluent Hypergeometric Function ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Volterra Type

In this article, we apply an integral transform-based technique to solve the fractional order Volterra-type integro-differential equation (FVIDE) involving the generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function in terms of several complex variables in the kernel. We also investigate and introduce the Elazki transform of Hilfer-derivative, generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function. In this article, we have established three results that are present in the form of lemmas, which give us new results on the above mentioned three functions, and by using these results we have derived our main results that are given in the form of theorems. Our main results are very general in nature, which gives us some new and known results as a particular case of results established here.

A System Associated with the Confluent Hypergeometric Function Φ3 and a Certain Linear Ordinary Differential Equation with Two Irregular Singular Points

1997 ◽  
Vol 08 (05) ◽  
pp. 689-702 ◽  
Author(s):  
Shun Shimomura
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Hypergeometric Function ◽  
Confluent Hypergeometric Function ◽  
Linear Ordinary Differential Equation ◽  
Singular Points ◽  
Third Order ◽  
Regular Type ◽  
Singular Loci ◽  
Irregular Singular Points

The confluent hypergeometric function Φ3 satisfies a system of partial differential equations on P1(C) × P1(C) with the singular loci x = 0, x = ∞, y = ∞ of irregular type and y = 0 of regular type. We obtain asymptotic expansions and Stokes multipliers of linearly independent solutions near the singular loci x = 0 and x = ∞. Applying the results we also clarify the global behaviour of the solutions of a third order linear ordinary differential equation with two irregular singular points.

scholarly journals A confluent hypergeometric integral equation

1982 ◽  
Vol 23 (1) ◽  
pp. 31-40 ◽  
Author(s):  
E. R. Love ◽  
T. R. Prabhakar ◽  
N. K. Kashyap
Keyword(s):  
Integral Equation ◽  
Hypergeometric Function ◽  
Integral Equations ◽  
Confluent Hypergeometric Function ◽  
Volterra Equations ◽  
Infinite Domain ◽  
Hypergeometric Integral ◽  
Image Position ◽  
Convolution Equations

Recently there have appeared papers ([7], [8]; also see [9]) in which integral equations with kernels involving the confluent hypergeometric functionhave been studied. These equations are mainly Volterra equations of the first kind except that they have infinite domain (0, ∞). The rest are of the related type with integrals over (x, ∞) instead of (0, x); and all are convolution equations.

The spectrum of the self-adjoint partial differential equation in any domain

1933 ◽  
Vol 29 (1) ◽  
pp. 23-44
Author(s):  
S. W. P. Steen
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Riemann Surface ◽  
General Solution ◽  
Boundary Conditions ◽  
Continuous Functions ◽  
The Self ◽  
Partial Differential ◽  
Image Position ◽  
Dimensional Domain

The object of this paper is to obtain the general solution to the self-adjoint partial differential equation in n dimensionswhere pij, q and ρ and bounded, continuous functions of (x1,…, xn) in a domain D and on its boundary, and where ∑pijXiXj≥0 for all (x1,…, xn) of D and all X1,…, Xn. The domain D is an n-dimensional domain and may be either the whole or part of a Riemann surface space of n dimensions. Its boundary is to consist of any number, zero, finite or enumerable, of continuous continua of n − 1 dimensions. These terms will be explained in paragraph II. The solution u = u(x1,…, xn; t) will be valid for (x1,…, xn) in D and t ≥ 0, and will satisfy boundary conditions of the type or similar, these conditions becoming identical at any part of the boundary of D that lies at infinity.

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