In mathematics the Mott polynomials ''s''_''n''(''x'') are polynomials introduced by who applied them to a problem in the theory of electrons. They are given by the exponential generating function :

e^=\sum_n s_n(x) t^n/n!.

Because the factor in the exponential has the power series :

\frac = -\sum_ C_k \left(\frac\right)^

in terms of

Catalan numbers In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Cata ...

C_k

, the coefficient in front of

x^k

of the polynomial can be written as :

s_n(x) =(-1)^k\frac\sum_C_C_\cdots C_

, according to the general formula for

generalized Appell polynomials In mathematics, a polynomial sequence \ has a generalized Appell representation if the generating function for the polynomials takes on a certain form: :K(z,w) = A(w)\Psi(zg(w)) = \sum_^\infty p_n(z) w^n where the generating function or kernel K(z ...

, where the sum is over all

compositions Composition or Compositions may refer to: Arts and literature * Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...

n=l_1+l_2+\cdots+l_k

n

into

k

positive odd integers. The empty product appearing for

k=n=0

equals 1. Special values, where all contributing Catalan numbers equal 1, are :

_n(x) = \frac.

_n(x) = \frac.

By differentiation the recurrence for the first derivative becomes :

s'(x) =- \sum_^ \frac C_k s_(x).

The first few of them are :

s_0(x)=1;

s_1(x)=-\fracx;

s_2(x)=\fracx^2;

s_3(x)=-\fracx-\fracx^3;

s_4(x)=\fracx^2+\fracx^4;

s_5(x)=-\fracx-\fracx^3-\fracx^5;

s_6(x)=\fracx^2+\fracx^4+\fracx^6;

The polynomials ''s''_''n''(''x'') form the associated

Sheffer sequence In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are ...

for –2''t''/(1–t²) . give an explicit expression for them in terms of the

generalized hypergeometric function In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which ...

₃F₀: :

s_n(x)=(-x/2)^n_3F_0(-n,\frac,1-\frac;;-\frac)

References

* * *{{Citation , last1=Roman , first1=Steven , title=The umbral calculus , url=https://books.google.com/books?id=JpHjkhFLfpgC , publisher=Academic Press Inc. arcourt Brace Jovanovich Publishers, location=London , series=Pure and Applied Mathematics , isbn=978-0-12-594380-2 , mr=741185 , id=Reprinted by Dover, 2005 , year=1984 , volume=111 Polynomials