Lie Theory and Special Functions Satisfying Second Order Nonhomogeneous Differential Equations

1970 ◽  
Vol 1 (2) ◽  
pp. 246-265 ◽  
Author(s):  
Willard Miller, Jr.
Keyword(s):  
Differential Equations ◽  
Special Functions ◽  
Second Order ◽  
Lie Theory

scholarly journals Fisher information of special functions and second-order differential equations

2008 ◽  
Vol 49 (8) ◽  
pp. 082104 ◽  
Author(s):  
R. J. Yáñez ◽  
P. Sánchez-Moreno ◽  
A. Zarzo ◽  
J. S. Dehesa
Keyword(s):  
Differential Equations ◽  
Fisher Information ◽  
Special Functions ◽  
Second Order ◽  
Second Order Differential Equations

scholarly journals On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials

2006 ◽  
Vol 58 (4) ◽  
pp. 726-767 ◽  
Author(s):  
Yik-Man Chiang ◽  
Mourad E. H. Ismail
Keyword(s):  
Differential Equations ◽  
Special Functions ◽  
Complete Characterization ◽  
Second Order ◽  
Value Distribution ◽  
Value Distribution Theory ◽  
Confluent Hypergeometric Functions ◽  
Number Of Zeros ◽  
Oscillation Of Solutions

AbstractWe show that the value distribution (complex oscillation) of solutions of certain periodic second order ordinary differential equations studied by Bank, Laine and Langley is closely related to confluent hypergeometric functions, Bessel functions and Bessel polynomials. As a result, we give a complete characterization of the zero-distribution in the sense of Nevanlinna theory of the solutions for two classes of the ODEs. Our approach uses special functions and their asymptotics. New results concerning finiteness of the number of zeros (finite-zeros) problem of Bessel and Coulomb wave functions with respect to the parameters are also obtained as a consequence. We demonstrate that the problem for the remaining class of ODEs not covered by the above “special function approach” can be described by a classical Heine problem for differential equations that admit polynomial solutions.

scholarly journals SOME RECURRENCE FORMULAS FOR A NEW CLASS OF SPECIAL POLYNOMIALS AND SPECIAL FUNCTIONS

2020 ◽  
Vol 14 (1) ◽  
Author(s):  
Snježana Maksimović ◽  
Nebojša Đurić ◽  
Ivan Vanja Boroja ◽  
Sandra Kosić-Jeremić
Keyword(s):  
Differential Equations ◽  
Special Functions ◽  
Second Order ◽  
Real Numbers ◽  
New Class ◽  
Recurrence Formulas ◽  
Special Polynomials ◽  
Summation Formulas ◽  
Integrable Functions ◽  
Square Integrable Functions

In this paper we used a new class of special functions and special polynomials which are solutions different Sturm Liouvile differential equations of second order. These functions form a basis of a space of square integrable functions over set of a real numbers. We investigated some properties of these polynomials and established some recurrence formulas. Using a new class of special functions, we obtained some useful summation formulas and recurrence formulas.

Convergent and asymptotic expansions of solutions of second-order differential equations with a large parameter

2014 ◽  
Vol 12 (05) ◽  
pp. 523-536 ◽  
Author(s):  
Chelo Ferreira ◽  
José L. López ◽  
Ester Pérez Sinusía
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Differential Equations ◽  
Initial Value Problem ◽  
Initial Conditions ◽  
Special Functions ◽  
Auxiliary Problem ◽  
Second Order ◽  
Large Parameter ◽  
Initial Value

We consider the second-order linear differential equation [Formula: see text] where x ∈ [0, X], X > 0, α ∈ (-∞, 2), Λ is a large complex parameter and g is a continuous function. The asymptotic method designed by Olver [Asymptotics and Special Functions (Academic Press, New York, 1974)] gives the Poincaré-type asymptotic expansion of two independent solutions of the equation in inverse powers of Λ. We add initial conditions to the differential equation and consider the corresponding initial value problem. By using the Green's function of an auxiliary problem, we transform the initial value problem into a Volterra integral equation of the second kind. Then using a fixed point theorem, we construct a sequence of functions that converges to the unique solution of the problem. This sequence has also the property of being an asymptotic expansion for large Λ (not of Poincaré-type) of the solution of the problem. Moreover, we show that the idea may be applied also to nonlinear differential equations with a large parameter.

scholarly journals On Generalisation of Polynomials in Complex Plane

2010 ◽  
Vol 2010 ◽  
pp. 1-9
Author(s):  
Maslina Darus ◽  
Rabha W. Ibrahim
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Complex Plane ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Second Order ◽  
Second Order Differential Equations

The generalised Bell and Laguerre polynomials of fractional-order in complex z-plane are defined. Some properties are studied. Moreover, we proved that these polynomials are univalent solutions for second order differential equations. Also, the Laguerre-type of some special functions are introduced.

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