Special functions

Special functions are some mathematical functions used in mathematical analysis or physics.Andrews, G. E., Askey, R., & Roy, R. (1999). Special functions (Vol. 71). Cambridge University Press.Silverman, R. A. (1972). Special functions and their applications. Courier Corporation.Nikiforov, A. F., & Uvarov, V. B. (1988). Special functions of mathematical physics (Vol. 205). Basel: Birkhäuser. Most of them appear in higher education. Some experts are studying numerical methods for them.Gil, A., Segura, J., & Temme, N. M. (2007). Numerical methods for special functions. Society for Industrial and Applied Mathematics.
Definition
In mathematics, most functions are defined as a solution of a differential equation. For example, the exponential function  is the solution of the ordinary differential equation . Due to this relation, some mathematicians are studying the connection between ODEs and special functions.Iwasaki, K., Kimura, H., Shimemura, S., & Yoshida, M. (2013). From Gauss to Painlevé: a modern theory of special functions (Vol. 16). Springer Science & Business Media.

Examples
 Gamma function, it is studied since EulerDavis, P. J. (1959). Leonhard euler's integral: A historical profile of the gamma function. The American Mathematical Monthly, 66(10), 849-869.Artin, E. (2015). The gamma function. Courier Dover Publications.
 Orthogonal polynomials, these are polynomials with special properties.Gautschi, W. (2004). Orthogonal polynomials. Oxford: Oxford University Press.Cohl, H. S., & Ismail, M. E. (Eds.). (2020). Lectures on Orthogonal Polynomials and Special Functions (Vol. 464). Cambridge University Press.Ismail, M., Ismail, M. E., & van Assche, W. (2005). Classical and quantum orthogonal polynomials in one variable (Vol. 13). Cambridge University Press.
 Matrix functions, these are studied in linear algebra and matrix analysis.Higham, N. J. (2008). Functions of matrices: theory and computation. Society for Industrial and Applied Mathematics.
For more examples, find textbooks named "special functions".

References

Other websites
National Institute of Standards and Technology, United States Department of Commerce. NIST Digital Library of Mathematical Functions. Archived from the original on December 13, 2018.

 Special functions at EqWorld: The World of Mathematical Equations
 Special functions and polynomials by Gerard 't Hooft and Stefan Nobbenhuis (April 8, 2013)
 Numerical Methods for Special Functions, by A. Gil, J. Segura, N.M. Temme (2007).
 R. Jagannathan, (P,Q)-Special Functions
 Specialfunctionswiki, a wiki about special functions

category:Functions and mappings