# Stieltjes polynomials

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##### 1: 31.15 Stieltjes Polynomials
###### §31.15(i) Definitions
Stieltjes polynomials are polynomial solutions of the Fuchsian equation (31.14.1). …
##### 2: 18.27 $q$-Hahn Class
###### §18.27(vi) Stieltjes–Wigert Polynomials
18.27.18 $S_{n}\left(x;q\right)=\sum_{\ell=0}^{n}\frac{q^{\ell^{2}}(-x)^{\ell}}{\left(q;% q\right)_{\ell}\left(q;q\right)_{n-\ell}}=\frac{1}{\left(q;q\right)_{n}}{{}_{1% }\phi_{1}}\left({q^{-n}\atop 0};q,-q^{n+1}x\right).$
18.27.19 $\int_{0}^{\infty}\frac{S_{n}\left(x;q\right)S_{m}\left(x;q\right)}{\left(-x,-% qx^{-1};q\right)_{\infty}}\mathrm{d}x=\frac{\ln\left(q^{-1}\right)}{q^{n}}% \frac{\left(q;q\right)_{\infty}}{\left(q;q\right)_{n}}\delta_{n,m},$
18.27.20 $\int_{0}^{\infty}S_{n}\left(q^{\frac{1}{2}}x;q\right)S_{m}\left(q^{\frac{1}{2}% }x;q\right)\exp\left(-\frac{(\ln x)^{2}}{2\ln\left(q^{-1}\right)}\right)% \mathrm{d}x=\frac{\sqrt{2\pi q^{-1}\ln\left(q^{-1}\right)}}{q^{n}\left(q;q% \right)_{n}}\delta_{n,m}.$
##### 3: 18.1 Notation
• Stieltjes–Wigert: $S_{n}\left(x;q\right)$.

• ##### 4: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes
18.29.2 $Q_{n}(z;a,b,c,d\mid q)\sim\frac{z^{n}\left(az^{-1},bz^{-1},cz^{-1},dz^{-1};q% \right)_{\infty}}{\left(z^{-2},bc,bd,cd;q\right)_{\infty}},$ $n\to\infty$; $z,a,b,c,d,q$ fixed.
For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). …
##### 5: Bibliography V
• H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.
• ##### 6: Bibliography W
• Z. Wang and R. Wong (2006) Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.
• ##### 7: Bibliography
• A. M. Al-Rashed and N. Zaheer (1985) Zeros of Stieltjes and Van Vleck polynomials and applications. J. Math. Anal. Appl. 110 (2), pp. 327–339.
• M. Alam (1979) Zeros of Stieltjes and Van Vleck polynomials. Trans. Amer. Math. Soc. 252, pp. 197–204.
• ##### 8: Bibliography K
• E. G. Kalnins and W. Miller (1993) Orthogonal Polynomials on $n$-spheres: Gegenbauer, Jacobi and Heun. In Topics in Polynomials of One and Several Variables and their Applications, pp. 299–322.
• T. H. Koornwinder (1977) The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.
• T. H. Koornwinder (1989) Meixner-Pollaczek polynomials and the Heisenberg algebra. J. Math. Phys. 30 (4), pp. 767–769.
• T. H. Koornwinder (2012) Askey-Wilson polynomial. Scholarpedia 7 (7), pp. 7761.
• T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.
• ##### 9: 25.2 Definition and Expansions
25.2.4 $\zeta\left(s\right)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma% _{n}(s-1)^{n},$
where the Stieltjes constants $\gamma_{n}$ are defined via
25.2.5 $\gamma_{n}=\lim_{m\to\infty}\left(\sum_{k=1}^{m}\frac{(\ln k)^{n}}{k}-\frac{(% \ln m)^{n+1}}{n+1}\right).$
25.2.10 $\zeta\left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0% pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{% 1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\mathrm{d}x,$ $\Re s>-2n$, $n=1,2,3,\dots$.
For $B_{2k}$ see §24.2(i), and for $\widetilde{B}_{n}\left(x\right)$ see §24.2(iii). …
##### 10: 25.6 Integer Arguments
###### §25.6(i) Function Values
where $\gamma_{1}$ is given by (25.2.5). …