# Stieltjes constants

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##### 1: 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 $\gamma_{n}=\lim_{m\to\infty}\left(\sum_{k=1}^{m}\frac{(\ln k)^{n}}{k}-\frac{(% \ln m)^{n+1}}{n+1}\right).$
##### 2: 25.6 Integer Arguments
where $\gamma_{1}$ is given by (25.2.5). …
##### 3: Errata
• Subsection 25.2(ii) Other Infinite Series

It is now mentioned that (25.2.5), defines the Stieltjes constants $\gamma_{n}$. Consequently, $\gamma_{n}$ in (25.2.4), (25.6.12) are now identified as the Stieltjes constants.

• ##### 4: 31.15 Stieltjes Polynomials
###### §31.15(ii) Zeros
This is the Stieltjes electrostatic interpretation. …
##### 5: 1.14 Integral Transforms
###### §1.14(vi) Stieltjes Transform
The Stieltjes transform of a real-valued function $f(t)$ is defined by … …
##### 6: 9.10 Integrals
9.10.15 $\int_{0}^{\infty}e^{-pt}\operatorname{Ai}\left(-t\right)\,\mathrm{d}t={\frac{1% }{3}e^{p^{3}/3}\left(\frac{\Gamma\left(\tfrac{1}{3},\tfrac{1}{3}p^{3}\right)}{% \Gamma\left(\tfrac{1}{3}\right)}+\frac{\Gamma\left(\tfrac{2}{3},\tfrac{1}{3}p^% {3}\right)}{\Gamma\left(\tfrac{2}{3}\right)}\right)},$ $\Re p>0$,
9.10.16 $\int_{0}^{\infty}e^{-pt}\operatorname{Bi}\left(-t\right)\,\mathrm{d}t={\frac{1% }{\sqrt{3}}e^{p^{3}/3}\left(\frac{\Gamma\left(\tfrac{2}{3},\tfrac{1}{3}p^{3}% \right)}{\Gamma\left(\tfrac{2}{3}\right)}-\frac{\Gamma\left(\tfrac{1}{3},% \tfrac{1}{3}p^{3}\right)}{\Gamma\left(\tfrac{1}{3}\right)}\right)},$ $\Re p>0$.
For the confluent hypergeometric function ${{}_{1}F_{1}}$ and the incomplete gamma function $\Gamma$ see §§13.1, 13.2, and 8.2(i). …
9.10.17 $\int_{0}^{\infty}t^{\alpha-1}\operatorname{Ai}\left(t\right)\,\mathrm{d}t=% \frac{\Gamma\left(\alpha\right)}{3^{(\alpha+2)/3}\Gamma\left(\tfrac{1}{3}% \alpha+\tfrac{2}{3}\right)},$ $\Re\alpha>0$.
##### 7: 18.1 Notation
• Racah: $R_{n}\left(x;\alpha,\beta,\gamma,\delta\right)$.

• Dual Hahn: $R_{n}\left(x;\gamma,\delta,N\right)$.

• Stieltjes–Wigert: $S_{n}\left(x;q\right)$.

• $q$-Racah: $R_{n}\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)$.

• Triangle: $P^{\alpha,\beta,\gamma}_{m,n}\left(x,y\right)$.

• ##### 8: 2.6 Distributional Methods
###### §2.6(ii) Stieltjes Transform
The Stieltjes transform of $f(t)$ is defined by … For a more detailed discussion of the derivation of asymptotic expansions of Stieltjes transforms by the distribution method, see McClure and Wong (1978) and Wong (1989, Chapter 6). Corresponding results for the generalized Stieltjes transform …An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). …
##### 9: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Let $X=[a,b]$ or $[a,b)$ or $(a,b]$ or $(a,b)$ be a (possibly infinite, or semi-infinite) interval in $\mathbb{R}$ . For a Lebesgue–Stieltjes measure $\,\mathrm{d}\alpha$ on $X$ let $L^{2}\left(X,\,\mathrm{d}\alpha\right)$ be the space of all Lebesgue–Stieltjes measurable complex-valued functions on $X$ which are square integrable with respect to $\,\mathrm{d}\alpha$ ,
1.18.13 $c_{n}=\left\langle f,\phi_{n}\right\rangle=\int_{a}^{b}f(x)\overline{\phi_{n}(% x)}\,\mathrm{d}x,$ $f\in L^{2}\left(X\right)$ ,
Note that the integral in (1.18.67) is not singular if approached separately from above, or below, the real axis: in fact analytic continuation from the upper half of the complex plane, across the cut, and onto higher Riemann Sheets can access complex poles with singularities at discrete energies $\lambda_{\mathrm{res}}-\mathrm{i}\Gamma_{\mathrm{res}}/2$ corresponding to quantum resonances, or decaying quantum states with lifetimes proportional to $1/\Gamma_{\mathrm{res}}$ . For this latter see Simon (1973), and Reinhardt (1982); wherein advantage is taken of the fact that although branch points are actual singularities of an analytic function, the location of the branch cuts are often at our disposal, as they are not singularities of the function, but simply arbitrary lines to keep a function single valued, and thus only singularities of a specific representation of that analytic function. This is accomplished by the variable change $x\to x{\mathrm{e}}^{\mathrm{i}\theta}$ , in $\mathcal{L}$ , which rotates the continuous spectrum $\boldsymbol{\sigma}_{c}\to\boldsymbol{\sigma}_{c}{\mathrm{e}}^{-2\mathrm{i}\theta}$ and the branch cut of (1.18.67) into the lower half complex plain by the angle $-2\theta$ , with respect to the unmoved branch point at $\lambda=0$ ; thus, providing access to resonances on the higher Riemann sheet should $\theta$ be large enough to expose them. This dilatation transformation, which does require analyticity of $q(x)$ , or an analytic approximation thereto, leaves the poles, corresponding to the discrete spectrum, invariant, as they are, as is the branch point, actual singularities of $\left\langle\left(z-T\right)^{-1}f,f\right\rangle$ . Let $T$ be a symmetric operator on a Hilbert space $V$ , so $\mathcal{D}(T)$ is dense in $V$ and $T\subset T^{**}\subset{T}^{*}$ . For $z\in\mathbb{C}\backslash\mathbb{R}$ let $N_{z}$ be the $z$ -eigenspace of ${T}^{*}$ , i.e., $N_{z}$ is the linear subspace of $\mathcal{D}({T}^{*})$ consisting of all $v$ for which ${T}^{*}v=zv$ . Then $\dim N_{z}$ is constant for $\Im z>0$ and also constant for $\Im z<0$ . Put $n_{+}=\dim N_{z}$ ( $\Im z>0$ ) and $n_{-}=\dim N_{z}$ ( $\Im z<0$ ), the deficiency indices for $T$ . Then $T$ has self-adjoint extensions iff $n_{+}=n_{-}$ .
##### 10: Bibliography C
• M. Carmignani and A. Tortorici Macaluso (1985) Calcolo delle funzioni speciali $\Gamma(x)$, $\log\Gamma(x)$, $\beta(x,y)$, $\operatorname{erf}(x)$, $\operatorname{erfc}(x)$ alle alte precisioni. Atti Accad. Sci. Lett. Arti Palermo Ser. (5) 2(1981/82) (1), pp. 7–25 (Italian).
• B. W. Char (1980) On Stieltjes’ continued fraction for the gamma function. Math. Comp. 34 (150), pp. 547–551.
• R. F. Christy and I. Duck (1961) $\gamma$ rays from an extranuclear direct capture process. Nuclear Physics 24 (1), pp. 89–101.
• T. Clausen (1828) Über die Fälle, wenn die Reihe von der Form $y=1+\frac{\alpha}{1}\cdot\frac{\beta}{\gamma}x+\frac{\alpha\cdot\alpha+1}{1% \cdot 2}\cdot\frac{\beta\cdot\beta+1}{\gamma\cdot\gamma+1}x^{2}+$ etc. ein Quadrat von der Form $z=1+\frac{\alpha^{\prime}}{1}\cdot\frac{\beta^{\prime}}{\gamma^{\prime}}\cdot% \frac{\delta^{\prime}}{\epsilon^{\prime}}x+\frac{\alpha^{\prime}\cdot\alpha^{% \prime}+1}{1\cdot 2}\cdot\frac{\beta^{\prime}\cdot\beta^{\prime}+1}{\gamma^{% \prime}\cdot\gamma^{\prime}+1}\cdot\frac{\delta^{\prime}\cdot\delta^{\prime}+1% }{\epsilon^{\prime}\cdot\epsilon^{\prime}+1}x^{2}+$ etc. hat. J. Reine Angew. Math. 3, pp. 89–91.