With the help of physical applications, we will develop more and more properties of these useful functions.
There is a HUGE amount of applications- which ones precisely depends on the area. Just to name a few classical references:
- R. Courant, D. Hilbert: Methods of Mathematical Physics
- F. Oberhettinger, W. Magnus: Anwendung der elliptischen Funktionen in Physik und Technik. (German)
- A. Wawrzynczyk, Group representations and special functions.
- N. Vilenkin, A. Klimyk, Representations of Lie groups and special functions,
3 volumes.
The literature on precise applications ranging from geodesy to electrodynamics, computer tomography and acoustics is so vast, that it can barely be overseen.
All great sources Ilka. I would add 4 more that are especially relevant: 2 general works and 2 pertaining to Bessel functions (widely applicable to diff eqns):
* LC Andrews (1985) Special function for engineers and applied mathematicians. Macmillan.
* ML Boas (1983) Mathematical methods in the physical sciences. Wiley.
* A Gray et al (1952) A treatise on Bessel functions and their applications to physics. Macmillan.
* CJ Tranter (1968) Bessel functions with some physical applications. Eng. Univ. Press.
As you said, the literature on application of special functions is enormous, but I think all of these sources are a great starting point.
More recent books related to this topic and applications are:
-G.E. Andrews, R.A. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.
- M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge, 2005.
- F.W.J. Olver and D.W. Lozier and R.F. Boisvert and Ch.W. Clark (eds.), NIST Handbook of Mathematical Functions. Cambridge University Press, 2010.
- N.M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley, 1996, ISBN: 0471-11313-1.
Recent issues relating special functions with Information Theory and Quantum Physics, together with open problems, can be seen in:
- J.S. Dehesa, A. Guerrero and P. Sanchez-Moreno, Information-theoretic-based spreading measures of orthogonal polynomials. Complex Analysis and Operator Theory 6(3) (2012) 585-601. 4.
- P. Sanchez-Moreno, A. Zarzo, J.S. Dehesa and A. Guerrero, Rényi entropy, Lp-norms and linearization of classical orthogonal polynomials. Applied Mathematics and Comput. 223 (2013) 25-33 5.
- J.S. Dehesa, A. Guerrero, J.L. Lopez and P. Sanchez-Moreno, Asymptotics (p ->∞) of Lp norms of hypergeometric ortogonal polynomials . J. Mathematical Chemistry 52 (2014) 283-300
- J.S. Dehesa, A. Martínez-Finkelshtein and J. Sánchez-Ruiz. Quantum information entropies and orthogonal polynomials Journal of Computational and Applied Mathematics 133 (2001) 23-46.
- A.I. Aptekarev, J.S. Dehesa, A. Martínez-Finkelshtein and R.J. Yáñez. Quantum expectation values of D-dimensional Rydberg states by use of Laguerre and Gegenbauer asymptotics J. Physics A 43 (2010) 145204
- A.I. Aptekarev, A. Martínez-Finkelshtein and J.S. Dehesa. Asymptotics of orthogonal polynomials entropy. J. Comp. Appl. Math. 233 (2010) 1355-1365.
- A.I. Aptekarev, J.S. Dehesa, A. Martínez-Finkelshtein and R.J. Yáñez. Discrete entropies of orthogonal polynomials. Constructive Approximation 30 (2009) 93-119
- J.S. Dehesa, R.J. Yáñez, A.I. Aptekarev and V. Buyarov . Strong asymptotics of Laguerre polynomials and information entropies of 2D harmonic oscillator and 1D Coulomb potentials. J. Mathematical Physics 39 (1998) 3050-3060.
- A.I. Aptekarev, V. Buyarov and J.S. Dehesa. Asymptotic behavior of Lp-norms and entropy for orthogonal polynomials . Russian Acad. of Sci. Sbornik Math. 185(8) (1994) 3-30; English trans- lation 82(2) (1995) 373-395.
- A. I. Aptekarev, J. S. Dehesa, P. Sánchez-Moreno and D. Tulyakov. Asymptotics of Lp-norms of Hermite polynomials and Rényi entropy of Rydberg oscillator states. Contemporary Mathematics 578 (2012) 19-29
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