Mott polynomials - Alchetron, The Free Social Encyclopedia

In mathematics the Mott polynomials s_n(x) are polynomials introduced by N. F. Mott (1932, p. 442) who applied them to a problem in the theory of electrons. They are given by the exponential generating function

e x ( 1 − t 2 − 1 ) / t = ∑ n s n ( x ) t n / n ! .

The first few of them are (sequence A137378 in the OEIS)

s 0 ( x ) = 1 ; s 1 ( x ) = − 1 2 x ; s 2 ( x ) = 1 4 x 2 ; s 3 ( x ) = − 3 4 x − 1 8 x 3 ; s 4 ( x ) = 3 2 x 2 + 1 16 x 4 ; s 5 ( x ) = − 15 2 x − 15 8 x 3 − 1 32 x 5 ; s 6 ( x ) = 225 8 x 2 + 15 8 x 4 + 1 64 x 6 ;

The polynomials s_n(x) form the associated Sheffer sequence for –2t/(1–t²) (Roman 1984, p.130). Arthur Erdélyi, Wilhelm Magnus, and Fritz Oberhettinger et al. (1955, p. 251) give an explicit expression for them in terms of the generalized hypergeometric function ₃F₀.

References

Mott polynomials Wikipedia

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