Department of Mathematics,
Faculty of Physics, MSU

Special functions of mathematical physics

The course "Special functions of mathematical physics" is a restored course, created by prof. A.F. Nikiforov. In this course a unified approach to constructing particular solutions to generalized hypergeometric equation is proposed. The generalized hypergeometric equation is an often used equation of mathematical and theoretical physics. The particular solutions are constructed using the generalized Rodrigue’s formula. The approach proposed is illustrated for several classical problems of mathematical and theoretical physics.   

The course content

1. The generalized equation of hypergeometric type and the algorithm of its simplification. Popynomials of hypergeometric type. An integrak form of the hypergeometric type functions.


2. Classical orthogonal polynomials. The Jacobi, Laguerre and Hermite polynomials. Behavior of the second solution of the differential equation for classical orthogonal polynomials. Sturm-Liouville problem. Completeness and closure of classical orthogonal polynomial system.


3. The examples of application of classical orthogonal polynomials in physics: energy levels and eigen functions of a linear harmonic oscillator; solution to the Schrodinger equation for a particle in a centrally symmetric field.


4. Classical orthogonal polynomials of discrete variable. A difference analogue to an equation of hypergeometric type. Difference Rodrigue's formula. Hahn, Chebyshev, Meixner, Kravchuk and Charlier polynomials.


5. The equation of hypergeometric type. Building of its particular solutions and functional relations for them.


6. The general properties of hypergeometric functions: recurrent relations, decomposition into power series. The choice of linearly independent solutions to an equation of hypergeometric type in case of different values of its parameters.


7. Solution of certain problems of mathematical physics and quantum mechanics.

Basic literature
Course Materials