An algebraic interpretation of the q-Meixner polynomials

Abstract

An algebraic interpretation of the q-Meixner polynomials is obtained. It is based on representations of ${\mathscr {U}}_q({\mathfrak {su}}(1,1))$ on q-oscillator states with the polynomials appearing as matrix elements of unitary q-pseudorotation operators. These operators are built from q-exponentials of the ${\mathscr {U}}_q({\mathfrak {su}}(1,1))$ generators. The orthogonality, recurrence relation, difference equation, and other properties of the q-Meixner polynomials are systematically obtained in the proposed framework.

Appendices

Appendix 1: “Dual” relations

The relations previously derived in the paper are usual relations found in the literature; however, they can be given a “dual” version by the process explained in Sect. 8. Here is the list of the relations that are obtained in this fashion.

Backward relation (6.7) $\rightarrow$

$$\begin{aligned} \theta ^2q^{x+1}&\left( 1-q^\beta \right) {\mathscr {M}}_{n}\left( q^{-(x+1)};q^{\beta -1},\theta ^2;q\right) \nonumber \\&=\theta ^2q^{x+1}\left( 1-q^{n+\beta }\right) {\mathscr {M}}_{n}\left( q^{-x};q^{\beta },\theta ^2;q\right) \nonumber \\&\quad -q\left( 1-q^n\right) \left( 1+\theta ^2q^{x+\beta }\right) {\mathscr {M}}_{n-1}\left( q^{-x};q^{\beta },\theta ^2q^{-1};q\right) \end{aligned}$$

(10.1)

Forward relation (6.10) $\rightarrow$

$$\begin{aligned} \frac{1-q^{-x}}{\theta ^2\left( 1-q^\beta \right) }&{\mathscr {M}}_{n}\left( q^{-(x-1)};q^{\beta -1},\theta ^2;q\right) \nonumber \\&={\mathscr {M}}_{n+1}\left( q^{-x};q^{\beta -1},\theta ^2q;q\right) -{\mathscr {M}}_{n}\left( q^{-x};q^{\beta -1},\theta ^2;q\right) \end{aligned}$$

(10.2)

Difference equation (6.13) $\rightarrow$

$$\begin{aligned} \left( 1-q^x\right)&{\mathscr {M}}_n\left( q^{-x};q^{\beta -1},\theta ^2;q\right) \nonumber \\&=-\left( 1-q^n\right) \left( 1+\theta ^2q^{x+\beta -1}\right) {\mathscr {M}}_{n-1}\left( q^{-x};q^{\beta -1},\theta ^2q^{-1};q\right) \nonumber \\&\quad +\left[ \left( 1-q^n\right) \left( 1+\theta ^2q^{x+\beta -1}\right) +\theta ^2q^x\left( 1-q^{n+\beta }\right) \right] \nonumber \\&\qquad \times {M}_n\left( q^{-x};q^{\beta -1},\theta ^2;q\right) \nonumber \\&\quad -\theta ^2q^x\left( 1-q^{n+\beta }\right) {\mathscr {M}}_{n+1}\left( q^{-x};q^{\beta -1},\theta ^2q;q\right) \end{aligned}$$

(10.3)

Complementary Backward relation (7.5) $\rightarrow$

$$\begin{aligned}&\frac{q}{\theta ^2}\frac{1-q^n}{1-q^{\beta -1}}{\mathscr {M}}_{n-1}\left( q^{-x};q^{\beta -1},\theta ^2q^{-1};q\right) \nonumber \\&\quad ={\mathscr {M}}_{n}\left( q^{-x};q^{\beta -2},\theta ^2;q\right) \nonumber \\&\qquad -{\mathscr {M}}_{n}\left( q^{-(x+1)};q^{\beta -2},\theta ^2q^{-1};q\right) \end{aligned}$$

(10.4)

Complementary Forward relation (7.7) $\rightarrow$

$$\begin{aligned}&\theta ^2q^x\left( 1-q^\beta \right) {\mathscr {M}}_{n+1}\left( q^{-x};q^{\beta -1},\theta ^2q;q\right) \nonumber \\&=\theta ^2\left( 1-q^{x+\beta }\right) {\mathscr {M}}_{n}\left( q^{-x};q^{\beta },\theta ^2;q\right) \nonumber \\&\quad -\left( q^n+\theta ^2\right) \left( 1-q^x\right) {\mathscr {M}}_{n}\left( q^{-(x-1)};q^{\beta },\theta ^2q;q\right) \end{aligned}$$

(10.5)

Recurrence relation (7.9) $\rightarrow$

$$\begin{aligned} q^{x+1}&\left( 1-q^{n}\right) {\mathscr {M}}_n\left( q^{-x};q^{\beta -1},\theta ^2;q\right) \nonumber \\&=-q\left( 1-q^x\right) \left( q^n+\theta ^2\right) {\mathscr {M}}_n\left( q^{-(x-1)};q^{\beta -1},\theta ^2q;q\right) \nonumber \\&\quad +\left[ q\left( 1-q^x\right) \left( q^n+\theta ^2\right) +\theta ^2\left( 1-q^{x+\beta }\right) \right] {\mathscr {M}}_n\left( q^{-x};q^{\beta -1},\theta ^2;q\right) \nonumber \\&\quad -\theta ^2\left( 1-q^{x+\beta }\right) {\mathscr {M}}_n\left( q^{-(x+1)};q^{\beta -1},\theta ^2q^{-1};q\right) \end{aligned}$$

(10.6)

Appendix 2: Useful q-series identities

A number of useful q-series identities are gathered here for convenience.

The q-binomial coefficients are defined as follows:

$$\begin{aligned} {n \brack k}_q =\frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}},\qquad k=0,1,2,\ldots ,n. \end{aligned}$$

(10.7)

They tend to the usual coefficients when $q\rightarrow 1$.

The little q-exponential, $e_q(z)$, and the big q-exponential, $E_q(z)$, are defined by

$$\begin{aligned} e_q(z)={}_{1}\phi _{0}\left( {\begin{array}{c} {0} \\ {-} \end{array}} \mid q;z \right) =\frac{1}{(z;q)_\infty },\qquad E_q(z)={}_{0}\phi _{0}\left( {\begin{array}{c} {-} \\ {-} \end{array}} \mid q;-z \right) =(-z;q)_\infty , \end{aligned}$$

(10.8)

for $|z|<1$. It is straightforward to see that $e_q(z)E_q(-z)=1$. From (10.8), one easily derives the following relations:

$$\begin{aligned}&e_q(\lambda q^n)=e_q(\lambda )(\lambda ;q)_n,&\qquad&e_q(\lambda q^{-n})=\frac{e_q(\lambda )}{(\lambda q^{-n};q)_n},\nonumber \\&E_q(\lambda q^n)=\frac{E_q(\lambda )}{(-\lambda ;q)_n},&\qquad&E_q(\lambda q^{-n})=E_q(\lambda )(-\lambda q^{-n};q)_n. \end{aligned}$$

(10.9)

The Baker–Campbell–Hausdorff formula admits two q-extensions [22, 23]. The first one is

$$\begin{aligned}&E_q(\lambda X)Ye_q(-\lambda q^\alpha X)=\sum _{n=0}^{\infty }\frac{\lambda ^n}{(q;q)_n}[X,Y]_n,\nonumber \\&[X,Y]_0=Y,\qquad [X,Y]_{n+1}=q^nX[X,Y]_n-q^\alpha [X,Y]_nX,\qquad n=0,1,2,\ldots \end{aligned}$$

(10.10)

The second one is

$$\begin{aligned}&e_q(\lambda X)YE_q(-\lambda q^\alpha X)=\sum _{n=0}^{\infty }\frac{\lambda ^n}{(q;q)_n}[X,Y]_n^\prime ,\nonumber \\&[X,Y]_0^\prime =Y,\qquad [X,Y]_{n+1}^\prime =X[X,Y]_n^\prime -q^{n+\alpha }[X,Y]_n^\prime X,\qquad n=0,1,2,\ldots \end{aligned}$$

(10.11)

Let us also record that for $XY=qYX$, one has

$$\begin{aligned} e_q(X+Y)=e_q(Y)e_q(X)\qquad \text {and}\qquad E_q(X+Y)=E_q(X)E_q(Y). \end{aligned}$$

(10.12)