Special Functions and Polynomials

Special functions are used in many applications of physics, engineering, and applied mathematics, such as electrodynamics, classical and modern physics, quantum mechanics, classical mechanics, and statistics, and, more recently, in the biological sciences and many other fields.

Furthermore, special polynomials are not only used in mathematical physics; they are also connected to deep problems of number theory.

You could say that every mathematical problem could be solved efficiently if the special functions that fit the solution were known. For example, certain equations that, like Airy's, have a symmetry connected to the third roots of the unity could be solved in a natural way by looking for the solution within particular expansions connected to the Chebyshev polynomials in two variables that have the same symmetry properties.

The hypergeometric functions constitute an important class within special functions that unify, through the introduction of appropriate parameters, most (if not all) parts of special functions, including elliptic integrals, beta functions, the incomplete Gamma function, Bessel functions, Legendre functions, classical orthogonal polynomials, Kummer confluent functions, and so on.

However, the broad range of special functions includes many other extensions and techniques, such as the multivariate generalizations of hypergeometric functions such as the Meijer G-function, the Lie and operational methods, and, in particular, G. Dattoli's monomiality principle.

The operational methods that can be found in earlier works of Euler and Lagrange, in relation to the construction of the generating functions of numerical sequences, were considered by G. Boole and O. Heaviside, who showed the connection between the derivative and the difference operator.

Their methods are at present included in the umbral calculus, a term introduced by J.J. Sylvester, since the exponent, for example that of $a^n$, is transformed into his ``shade'', which appears in $a_n$, so that powers are considered as sequences. The operational methods of umbral calculus, according to the notation of E. Lucas, have been made rigorous by G-C. Rota and S.M. Roman.

• special functions
• matrix functions
• eigenvalues
• differential and integral equations
• number theory