A. Zhedanov, 9J-SYMBOLS OF THE OSCILLATOR ALGEBRA AND KRAWTCHOUK POLYNOMIALS IN 2 VARIABLES, Journal of physics. A, mathematical and general, 30(23), 1997, pp. 8337-8353
A simple generating function for the 9j-symbols of the oscillator alge
bra is found. On the basis of this function it is shown that correspon
ding 9j-symbols are expressed in terms of polynomials Q(pr)(k, m) in t
wo discrete variables which are orthogonal with respect to the trinomi
al distribution. These polynomials depend on four independent paramete
rs and can be considered as a two-dimensional analogue of the Krawtcho
uk polynomials. Difference-difference relations, the factorization cha
in, the duality property, and the Rodriguez formula for these polynomi
als are obtained. It is shown that the polynomials Q(pr)(k, m) are eig
enfunctions of two commuting difference operators. These polynomials a
re also covariant with respect to two commuting difference derivation
operators. In a special symmetric case these polynomials admit a simpl
e factorized expression in terms of two distinct ordinary Krawtchouk p
olynomials. In another special case we obtain an explicit expression o
f 9j-symbols in terms of the Appell hypergeometric function F-1 in two
variables.