Yu. A. Brychkov

On some formulas for the Appell function F3 (a,a′, b, b′; c; w,z)

New relations and transformation formulas for the Appell function and the confluent Appell functions (Humbert functions) are obtained. These relations include limit formulas, integral representations, differentiation formulas. Various finite and infinite summation formulas are also derived.

On the derivatives of the Bessel and Struve functions with respect to the order

Closed form expressions are obtained for the first derivatives with respect to the order of the Bessel functions J ν(z), Y ν(z), I ν(z), K ν(z); integral Bessel functions Ji ν(z), Yi ν(z), Ki ν(z); and Struve functions Hν(z), Lν(z) at ν = ±n, ν = ±n + 1/2, with n = 0, 1, 2….

On some properties of the Marcum Q function

A closed expression for Q ν(a, b) with integer ν in terms of a confluent Appell function, differentiation formulas with respect to a and b, generating functions and other relations are given.

On multiple sums of special functions

Sums of the form are obtained for various special functions f ν(z).

On some properties of the Nuttall function Qμ, ν(a, b)

Differentiation formulas for the Nuttall function Qμ, ν(a, b) with respect to a and b, generating functions, a closed expression with integer ν in terms of a confluent Appell function, and other relations are given.

On some properties of the generalized Bernoulli and Euler polynomials

A closed expression, recurrency relation, differentiation formulas and other new relations for the generalized Bernoulli and Euler polynomials are given. Formulas of summation containing the generalized Bernoulli and Euler polynomials and various special functions, are derived.

Higher derivatives of the Bessel functions with respect to the order

ABSTRACT Closed expressions are given for the first derivatives with respect to the order of the Bessel functions , Neumann function , Macdonald function and Kelvin functions for any value ν, and for the second and third derivatives at integer points.

On the derivatives of the Legendre functions and with respect to μ and ν

Formulas of derivatives of the Legendre functions and with respect to μ and ν at μ, ν=0,±1,±2, … are given.

Power expansions of powers of trigonometric functions and series containing Bernoulli and Euler polynomials

Differentiation formulas and power expansions are derived for sin n  z,  cos n  z,  tan n  z and cot n  z. Summation formulas containing Bernoulli and Euler polynomials and numbers are obtained.

Two series containing the Laguerre polynomials

Formulas of summation of the series are obtained for any integer n.