--- title: 'Stieltjes Functions: Theory & Applications' url: https://www.emergentmind.com/topics/stieltjes-functions type: topic --- # Stieltjes Functions: Theory & Applications Stieltjes functions, in the classical sense, are functions on \((0,\infty)\) or on a slit complex plane that admit a Cauchy-type integral representation against a positive measure, typically \[ f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0, \] with the integral convergent for \(x>0\). They form a distinguished cone at the intersection of complete monotonicity, Laplace transform theory, Pick–Nevanlinna theory, moment problems, Bernstein functions, and special-function analysis. Modern developments enlarge this picture in several directions: generalized Stieltjes classes \(S_\alpha\) of arbitrary order \(\alpha>0\), differential and finite-order characterizations, exact-order invariants, matrix-valued analogues, and applications to hypergeometric, gamma, and Lambert \(W\) functions, as well as to ratios of entire functions and moment problems [1111.4271][1706.00606][1506.01600]. ## 1. Classical definition and analytic characterizations A classical Stieltjes function is usually defined by the representation \[ f(z)=a+\int_0^\infty \frac{d\mu(t)}{z+t}, \] or, in equivalent real-variable form, \[ f(x)=c+\int_{[0,\infty)}\frac{1}{x+t}\,p(dt),\qquad x>0, \] where the constant is nonnegative and the representing measure is positive [1706.00606][1705.02493]. In the generalized theory developed by Sokal, Berg, and others, the classical class is denoted \(S_1\), emphasizing that it is the order-\(1\) case of a larger hierarchy [1111.4271]. Several complex-analytic characterizations are standard. One form states that a function \(f:(0,\infty)\to\mathbb R\) is Stieltjes if and only if it extends holomorphically to \(\mathbb C\setminus(-\infty,0]\), is nonnegative on \((0,\infty)\), and satisfies \[ \Im f(z)\le 0 \qquad \text{for }\Im z>0 \] [1103.5640]. Closely related slit-plane classes \(\mathbf S\) and \(\mathbf S^{-1}\) are defined by the sign of the imaginary part in the upper half-plane together with positivity on the positive axis; they admit integral representations with kernels \((\lambda+z)^{-1}\) and \(z(\lambda+z)^{-1}\), respectively [1103.0118]. The classical class is tightly linked to complete monotonicity. A function \(f\) is completely monotone on \((0,\infty)\) if \[ (-1)^n f^{(n)}(x)\ge 0,\qquad n\ge 0,\ x>0. \] By Bernstein’s theorem, this is equivalent to \(f\) being a Laplace transform of a positive measure [1706.00606]. Every Stieltjes function is completely monotone, and Widder’s real-variable characterization identifies exactly which completely monotone functions are Stieltjes: \(f\) is Stieltjes if and only if \((x^k f(x))^{(k)}\) is completely monotone for every \(k\ge 0\) [1706.00606]. This criterion remains one of the central structural tools in the subject. ## 2. Generalized classes \(S_\alpha\) and exact Stieltjes order For \(\alpha>0\), a generalized Stieltjes function of order \(\alpha\) is a function of the form \[ f(x)=\int_{[0,\infty)}\frac{d\mu(t)}{(x+t)^\alpha}+c, \qquad c\ge 0, \] with \(\mu\) positive and the integral convergent; the corresponding class is denoted \(S_\alpha\) [1905.04131]. The same class admits a Laplace-transform representation \[ f(x)=\frac{1}{\Gamma(\alpha)}\int_0^\infty e^{-xt}t^{\alpha-1}K(t)\,dt+c, \] where \(K\) is completely monotone [1905.04131]. This makes explicit that generalized Stieltjes functions are Laplace transforms of \(t^{\alpha-1}\) multiplied by a completely monotone kernel. A basic inclusion holds: \[ S_{\alpha_1}\subset S_{\alpha_2}\qquad \text{if }0<\alpha_1<\alpha_2 \] [1111.4271]. The order parameter is not merely formal. The paper "Generalized Stieltjes transforms: basic aspects" defines the exact Stieltjes order \[ \alpha^*[f]:=\inf\{\alpha>0:\ f\in S_\alpha\}, \] proves that this infimum is attained, and gives a criterion for exactness in terms of monotonicity of a fractional integral \(\Phi_\varepsilon\) built from a representing measure at a larger order [1111.4271]. Compact support of the representing measure forces exactness, and hypergeometric examples show that nontrivial special functions can have explicitly computable exact order [1111.4271]. Within the generalized hierarchy, the class \(S_2\) is distinguished. A function \(f:(0,\infty)\to(0,\infty)\) is logarithmically completely monotone if \(-(\log f)'\) is completely monotone. Horn’s theorem characterizes this by complete monotonicity of all positive powers \(f^a\). A central result used repeatedly in special-function applications is \[ S_2\subset \mathcal L, \] where \(\mathcal L\) denotes the class of logarithmically completely monotone functions [1905.04131]. The same source stresses that this inclusion is special to order \(2\): for \(\alpha>2\), there exist functions in \(S_\alpha\setminus\mathcal L\) [1905.04131]. This distinguishes order \(2\) from the rest of the generalized scale. ## 3. Differential tests, finite-order truncations, and real-variable structure The generalized order-\(\lambda\) theory has a precise real-variable characterization. Sokal introduced differential operators \[ T_{n,k}(f)(x)=(-1)^n x^{-(n+1-\lambda)}\bigl(x^{k+n+1-\lambda}f^{(n)}(x)\bigr)^{(k)}, \] and proved that, for \(f\in C^\infty((0,\infty))\), the condition \[ T_{n,k}(f)(x)\ge 0 \quad \forall n,k\ge 0,\ x>0 \] is equivalent to \(f\) being a generalized Stieltjes function of order \(\lambda\) [1706.00606]. Koumandos and Pedersen simplified this by introducing \[ c_k(f)(x):=x^{1-\lambda}\bigl(x^{\lambda-1+k}f(x)\bigr)^{(k)}, \] and proving \[ T_{n,k}(f)(x)=(-1)^n(c_k(f))^{(n)}(x). \] Hence \(f\) is generalized Stieltjes of order \(\lambda\) if and only if every \(c_k(f)\) is completely monotone [1706.00606]. This reformulation compresses Sokal’s two-parameter inequality array into one family of complete-monotonicity conditions. The same paper studies finite truncations. For fixed \(N\), \[ C_N=\{f\in C^\infty((0,\infty)):\ c_k(f)\ \text{is completely monotone for }k=0,\dots,N\}, \] and \(C_\infty=\bigcap_{N\ge 0}C_N\) is exactly the generalized Stieltjes class of order \(\lambda\) [1706.00606]. Membership in \(C_N\) is characterized by positivity properties of derived measures \[ \mu_k:=(-1)^k s^k\frac{d^k}{ds^k}(s^{-\lambda}\mu) \] in the distribution sense, where \(\mu\) is the representing measure in a Laplace form \[ f(x)=c+\int_0^\infty e^{-xs}s^{\lambda-1}\,d\mu(s) \] [1706.00606]. This finite-order theory shows that requiring only finitely many complete-monotonicity conditions imposes a truncated hierarchy of measure-theoretic positivity constraints rather than full generalized Stieltjes structure. A different finite-order program appears in "Stieltjes functions of finite order and hyperbolic monotonicity" [1604.05267]. There, \(S_k\) denotes the class of nonnegative functions satisfying Widder-type inequalities only up to order \(k\), interpreted in the measure sense: \[ (-1)^{n-1}(xf(x))^{(2n-1)}\ge 0,\qquad n=1,\dots,k. \] These functions need not be smooth. For \(k\ge 2\), they admit a unique representation \[ f(x)=a+\int_0^\infty \Phi_{k-1}(x,t)\,\mu_k(dt), \] where \(\Phi_{k-1}\) is a nonnegative truncated Laurent kernel converging pointwise to \((x+t)^{-1}\) as \(k\to\infty\) [1604.05267]. Two finite-order notions therefore coexist in the literature: truncated \(c_k\)-complete-monotonicity in the generalized \(S_\lambda\) framework, and truncated Widder inequalities in the classical \(S\)-framework. ## 4. Special functions, transcendental equations, and zero-free phenomena A large modern literature identifies explicit special functions as members of Stieltjes classes. For generalized hypergeometric functions, several parameter regimes imply that \({}_{p+1}F_p(a;b;-x)\) is a Stieltjes function, while Meijer \(G\)-based Laplace representations yield complete monotonicity for related \({}_pF_q\) and \({}_{p+1}F_p\) families [1705.02493]. These representations are then used to compute branch-cut jumps and average values, obtain generalized Stieltjes formulas with kernels \((z^\alpha+y^\alpha)^{-1}\), and derive univalence and distortion inequalities in the half-plane \(\Re z<1\) [1705.02493]. The order-\(2\) class is especially rich in explicit examples. Nielsen’s beta function \[ \beta(z)=\sum_{n=0}^\infty \frac{(-1)^n}{z+n} \] is shown to belong to \(S_2\), hence to \(\mathcal L\), and this is extended to broad alternating series families, ratios of Gamma functions, Prym’s function, and remainders in asymptotic expansions of Barnes’ double Gamma function [1905.04131]. The same source proves, for instance, that \[ \log\frac{\Gamma(x)\Gamma(x+a+b)}{\Gamma(x+a)\Gamma(x+b)}\in S_2 \] for \(a,b>0\), while \[ \frac{\Gamma(x)}{\Gamma(x+N+s)}\in S_{N+1} \] for \(N\in\{0,1,2,\dots\}\) and \(00\) for \(x>0\). For shift arrays \(a_1\le\cdots\le a_n\) and \(b_1\le\cdots\le b_n\), the ratio \[ W_f(x)=\frac{\prod_{k=1}^n f(x+a_k)}{\prod_{k=1}^n f(x+b_k)} \] has logarithmic derivatives that become generalized Stieltjes functions under weak supermajorization and power-sum cancellation conditions [2107.00905]. In particular, \[ (-1)^{p-1}\partial_z^p\log W_f(z) \] is generalized Stieltjes of order \(p+2\), and under \(\sum a_k=\sum b_k\), \[ (-1)^p\partial_z^{p-1}\log W_f(z) \] is generalized Stieltjes of order \(p+1\) [2107.00905]. Applications include Euler’s gamma function, Barnes’ \(G\)-function, and multiple gamma functions, as well as an explicit link to Prouhet–Tarry–Escott identities through the same power-sum constraints [2107.00905]. A different construction connects Stieltjes classes to zero localization of entire functions. If \(\psi\) belongs to the slit-plane Stieltjes class \(\mathbf S\), then \[ E_\psi(z)=\sum_{k=0}^{\infty}\frac{\psi(k+1)}{k!}z^k \] is a Hurwitz stable entire function when \(\psi\) is generic; similarly, if \(\psi\in\mathbf S^{-1}\), then \[ E_\psi^{-}(z)=\sum_{k=0}^{\infty}\frac{\psi(k)}{k!}z^k \] is Hurwitz stable unless \(\psi(0)=0\), in which case \(z^{-1}E_\psi^{-}(z)\) is Hurwitz stable [1103.0118]. The same paper extends the construction by replacing \(e^z\) with an entire function from the Laguerre–Pólya class of type I, thereby generating broad families of Hurwitz stable entire functions from Stieltjes data [1103.0118]. Matrix-valued Stieltjes functions furnish the natural noncommutative generalization. For a real \(\alpha\), the class \(\mathcal S_{q;[\alpha,\infty)}\) consists of \(q\times q\) matrix-valued holomorphic functions on \(\mathbb C\setminus[\alpha,\infty)\) with positive semidefinite imaginary part on \(\mathbb C^+\) and positive semidefinite values on \((-\infty,\alpha)\) [1506.01600]. These functions admit integral representations such as \[ F(z)=\gamma+\int_{[\alpha,\infty)}\frac{1+t-\alpha}{t-z}\,\mu(dt) \] and, for a distinguished subclass, \[ F(z)=\int_{[\alpha,\infty)}\frac{1}{t-z}\,\sigma(dt) \] with non-negative Hermitian matrix measures [1506.01600]. The framework is closely tied to truncated matricial Stieltjes moment problems, and it is stable under a Moore–Penrose inverse transformation: \[ G(z)=-\frac{1}{z-\alpha}F(z)^\dagger\in \mathcal S_{q;[\alpha,\infty)} \] whenever \(F\in \mathcal S_{q;[\alpha,\infty)}\) [1506.01600]. Range and kernel of \(F(z)\) are determined entirely by the measure parameters, which is crucial for matricial moment theory [1506.01600]. ## 6. Finite-horizon, differential-calculus, and alternative usages of the term The term “Stieltjes functions” also appears in a distinct real-variable literature on Stieltjes differential calculus. There the central datum is a derivator \(g\), usually nondecreasing and left-continuous, which induces a Stieltjes measure \(\mu_g\), a \(g\)-topology, a Stieltjes derivative \(f'_g\), and function spaces such as \(C_g\), \(BC_g\), \(W^{1,p}_g\), and \(\mathcal{BD}_g^k\) [2211.07279][2503.23962]. In this setting, the phrase “Stieltjes function spaces” refers to spaces of functions defined or differentiated with respect to \(g\), not to the classical holomorphic Stieltjes-transform class \(S_\alpha\) [2211.07279]. This alternative usage has developed its own compactness and approximation theory. Compactness criteria of Ascoli–Arzelà and Kolmogorov–Riesz type are proved for \(BC_g\), \(BUC_g\), and \(L^p_g\); Stieltjes–Sobolev spaces \(W^{1,p}_g\) are shown to be Banach, reflexive for \(1