Structure of the Xp Matrices in the Simple Harmonic Oscillator Representation

The m, n element of X^p in the one‐dimensional simple harmonic oscillator representation is shown to be in the form $${\mathrm{〈m\; |\; x}}^p{\mathrm{|\; n〉=\beta }}^{\text{p|2}}{\text{[(N−r+1)(N−r+3)(N−r+5)···(N+r−1)]}}^{\frac{1}{2}}\left(\begin{array}{c}p\\ \frac{1}{2}\mathrm{(p-r)}\end{array}\right)2^{\text{−p/2}}{F}_r^p\mathrm{(N),}$$ where N=m+n+1, r=m—n, β=ℏ/2μω and F_r^p(N) is a simple polynomial in N, purely even or odd and with small, positive, integral coefficients. A number of simple recursion relations are developed for the F polynomials.