TWO NORMAL ORDERING PROBLEMS AND CERTAIN SHEFFER POLYNOMIALS

The first normal ordering problem involves bosonic harmonic oscillator creation and annihilation operators (Heisenberg algebra). It is related to the problem of finding the finite transformation generated by L_k−1 := −z^k ∂_z, k ∈ ℤ, z ∈ ℂ (conformal algebra generators). It can be formulated in terms of a subclass of Sheffer polynomials called Jabotinsky polynomials. The coefficients of these polynomials furnish generalized Stirling number triangles of the second kind, called S2(k;n,m) for k ∈ ℤ. Generalized Stirling-numbers of the first kind, S1(k;n, m) are also defined.

The second normal ordering problem appears in thermo-field dynamics for the harmonic Bose oscillator. Again Sheffer polynomials appear. They relate to Euler numbers and iterated sums of squares. In a different approach to this problem one solves the differential-difference equation f_n+1 = f'_n + n² f_n-1, n > = 1, with certain inputs f₀ and f₁ = f'₀.

In this case the integer coefficients of the special Sheffer polynomials which emerge have an interpretation as sum over multinomials for some subset of partitions.