§30.16 Methods of Computation

For small $|{\gamma }^2|$ we can use the power-series expansion (30.3.8). Schäfke and Groh (1962) gives corresponding error bounds. If $|{\gamma }^2|$ is large we can use the asymptotic expansions in §30.9. Approximations to eigenvalues can be improved by using the continued-fraction equations from §30.3(iii) and §30.8; see Bouwkamp (1947) and Meixner and Schäfke (1954, §3.93).

Another method is as follows. Let $n-m$ be even. For $d$ sufficiently large, construct the $d\times d$ tridiagonal matrix $𝐀=[A_{j,k}]$ with nonzero elements

and real eigenvalues ${\alpha }_{1,d}$, ${\alpha }_{2,d}$, $\mathrm{\dots }$, ${\alpha }_{d,d}$, arranged in ascending order of magnitude. Then

and

The eigenvalues of $𝐀$ can be computed by methods indicated in §§3.2(vi), 3.2(vii). The error satisfies

If $|{\gamma }^2|$ is large, then we can use the asymptotic expansions referred to in §30.9 to approximate ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$.

If ${\lambda }_n^m\left({\gamma }^2\right)$ is known, then we can compute ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ (not normalized) by solving the differential equation (30.2.1) numerically with initial conditions $w(0)=1$, $w^{\prime }(0)=0$ if $n-m$ is even, or $w(0)=0$, $w^{\prime }(0)=1$ if $n-m$ is odd.

If ${\lambda }_n^m\left({\gamma }^2\right)$ is known, then ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ can be found by summing (30.8.1). The coefficients $a_{n,r}^m({\gamma }^2)$ are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5).

A fourth method, based on the expansion (30.8.1), is as follows. Let $𝐀$ be the $d\times d$ matrix given by (30.16.1) if $n-m$ is even, or by (30.16.6) if $n-m$ is odd. Form the eigenvector ${[e_{1,d},e_{2,d},\mathrm{\dots },e_{d,d}]}^{\mathrm{T}}$ of $𝐀$ associated with the eigenvalue ${\alpha }_{p,d}$, $p=\lfloor \frac{1}{2}(n-m)\rfloor +1$, normalized according to

Then

For error estimates see Volkmer (2004a).

The coefficients $a_{n,k}^m({\gamma }^2)$ calculated in §30.16(ii) can be used to compute $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$ from (30.11.3) as well as the connection coefficients $K_n^m(\gamma )$ from (30.11.10) and (30.11.11).

For other methods see Van Buren and Boisvert (2002, 2004).