--- title: Truncated Mellin Transforms url: https://www.emergentmind.com/topics/truncated-mellin-transforms type: topic --- # Truncated Mellin Transforms Searching arXiv for relevant papers on truncated Mellin transforms and truncated Mellin moments. Searching arXiv for "truncated Mellin transform" and "truncated Mellin moments". The expression *truncated Mellin transform* is used in several adjacent but nonidentical senses. In the classical one-sided setting, the Mellin transform is \[ M\{f(t);s\}=\int_{0}^{\infty} t^{\,s-1}f(t)\,dt, \] and truncation may mean either that the integrand is built from a terminating special-function series or that the function being transformed is supported only on a proper subinterval such as \((0,1)\). In perturbative QCD, the closely related notion of a truncated Mellin moment replaces the full integral over \(x\in[0,1]\) by an integral over an experimentally accessible interval, typically \([x_0,1]\) or \([x_{\min},x_{\max}]\). Across these literatures, the Mellin kernel is retained while the transformed object is restricted in a way that alters convergence, analytic continuation, and evolution properties [1906.08057] [1405.5315] [2601.17092]. ## 1. Terminological scope and basic objects A common ambiguity in the literature is terminological: in special-function analysis, truncation usually refers to termination of a hypergeometric series or compact support of the transformed function, whereas in QCD it refers to restriction of the integration range in Bjorken-\(x\). The constructions are related by Mellin methodology but are not interchangeable. | Context | Representative object | Primary role | |---|---|---| | Special functions | \(M\{f(t);s\}=\int_0^\infty t^{s-1}f(t)\,dt\) with a terminating or truncated hypergeometric factor | Closed forms, summation theorems, meromorphic continuation | | Compact support on \((0,1)\) | \(\int_0^1 x^{s-1}f(x)\,dx\) | Endpoint analysis, contour methods, arithmetic applications | | Perturbative QCD | \(M_n(x_0,Q^2)=\int_{x_0}^1 x^{n-1}f(x,Q^2)\,dx\) | Evolution in the measurable \(x\)-range | In QCD notation, the full Mellin moment of a parton density or structure function is \[ M^n(Q^2)=\int_0^1 dx\,x^{n-1}f(x,Q^2), \] while the truncated counterpart is \[ M_n(x_0,Q^2)=\int_{x_0}^1 dx\,x^{n-1}f(x,Q^2). \] When \(x_0\to 0\), the usual full Mellin moment is recovered. In the compact-support setting, the analogous restriction is intrinsic rather than imposed by measurement: functions such as \(1/\operatorname{arctanh}x\) and \(1/(\sqrt{1-x^2}\,\operatorname{arctanh}x)\) are treated as supported on \((0,1)\), and their Mellin transforms converge absolutely for \(\Re s>1\) [1405.5315] [1607.08397] [2601.17092]. ## 2. Terminating hypergeometric constructions A distinctly classical use of truncated Mellin transforms appears in the analysis of terminating and truncated Clausen hypergeometric series with unit argument. Qureshi, Jabee, and Ahamad derive new summation theorems for \({}_3F_2(\cdots;1)\) when one upper and one lower parameter is a negative integer, and then use these theorems to obtain Mellin transforms of the product of an exponential function and Goursat’s truncated hypergeometric function [1906.08057]. The starting point is the one-sided Mellin transform with the convergence conditions \(\Re(s)>0\) and, in the exponential case, \(\Re(u)>0\). For the truncated Goursat function, term-by-term integration yields the fundamental identity \[ M\Bigl\{e^{-u t}\,{}_{2}F_{2}(-m,a;\,-m-\ell,b;\,u t)\;;\,s\Bigr\} = u^{-s}\,\Gamma(s)\; {}_{3}F_{2}\bigl(-m,\;a,\;s;\;-m-\ell,\;b;\;1\bigr). \] The transform problem is thereby reduced to a terminating \({}_3F_2(1)\) evaluation. The paper organizes the relevant reductions through sixteen summation theorems, including truncated forms of Watson’s, Saalschütz’s, Whipple’s, and Dixon’s theorems. The operational recipe is uniform: begin with term-by-term integration, recognize the emergent \({}_3F_2(1)\), apply the appropriate truncated summation theorem, and rewrite the result as a ratio of Gamma or shifted Pochhammer factors. The final closed forms are listed in sixteen cases, Eqs. (3.2)–(3.17). Parameter restrictions have the usual Mellin-transform and hypergeometric character: \(\Re(s)>0\), \(\Re(u)>0\), nonnegative truncation indices, and exclusion of poles arising when a Pochhammer denominator vanishes. The resulting Gamma-ratio expressions are meromorphic in the parameters and agree with the analytic continuation of \({}_2F_2\) [1906.08057]. This use of truncation is algebraic rather than geometric. The series is finite from the outset, so interchange of sum and integral is justified by uniform convergence of the finite sum, and the Mellin transform becomes an efficient device for converting hypergeometric termination into explicit closed forms. ## 3. Compactly supported Mellin transforms and contour methods A second special-function usage concerns Mellin transforms of compactly supported functions on \((0,1)\). The transforms \[ M_1(s)=\int_0^1 x^{s-1}\,\frac{1}{\operatorname{arctanh}(x)}\,dx, \qquad M_2(s)=\int_0^1 x^{s-1}\,\frac{1}{\sqrt{1-x^2}\,\operatorname{arctanh}(x)}\,dx \] are studied as Mellin transforms of functions supported on \((0,1)\), with absolute convergence for \(\Re s>1\) [2601.17092]. Using Taylor expansions on \(|x|<1\), one obtains Beta-function representations and then Gamma-series expansions. A convenient form is \[ \Phi_2(s) = \pi\,2^{\,1-s} \sum_{n=0}^{\infty} p_n\,4^{-n}\, \frac{\Gamma(s+2n-1)} {\Gamma\!\bigl(\tfrac{s+2n}{2}\bigr)^2}, \] and \[ \Phi_1(s) = \pi\,2^{\,1-s} \sum_{n=0}^{\infty} q_n\,4^{-n}\, \frac{\Gamma(s+2n-1)} {(s+2n)\,\Gamma\!\bigl(\tfrac{s+2n}{2}\bigr)^2}, \] valid for \(\Re s>1\). For even integers \(s=2m\), these specialize to rational binomial-coefficient expansions. At odd integers \(s=2N+1\), the substitution \(x=\tanh u\) converts the Mellin integrals into hyperbolic integrals with a \(1/u\) factor. After binomial expansion and use of the Frullani integral, \[ \int_0^\infty u^{-1}\bigl(e^{-\alpha u}-e^{-\beta u}\bigr)\,du=\ln(\beta/\alpha), \] the transforms are expressed as finite rational combinations of derivatives of the Dirichlet eta and beta functions: \[ \Phi_1(2N+1)=\sum_{i=0}^{N} A_{i,N}\,\eta'(-2i-1), \qquad \Phi_2(2N+1)=\sum_{i=0}^{N} B_{i,N}\,\beta'(-2i). \] Even-integer values require a different method because the relevant improper integrals cannot be dispatched by parity. The contour-integral framework introduces \[ I_{q,n} = \int_0^\infty \frac{\sinh^{2q+1}z\,\ln z}{\cosh^n z}\,dz, \] shows that the vertical sides of a rectangular contour vanish, and evaluates the boundary integral by residues at the poles \[ z_\ell=\ln\frac{(2\ell+1)\pi}{2}+i\pi/2, \qquad \ell=0,1,2,\dots. \] A bridge identity then relates the original Mellin-type integrals with \(1/z\) factors to these logarithmic hyperbolic integrals: \[ \int_0^\infty\frac{\sinh^{2q}z}{z\,\cosh^n z}\,dz = -2q\int_0^\infty \ln z\;\frac{\sinh^{2q-1}z}{\cosh^{n-1}z}\,dz + n\int_0^\infty \ln z\;\frac{\sinh^{2q+1}z}{\cosh^{n+1}z}\,dz. \] The resulting closed forms involve \(\zeta'(2p+2)/\pi^{2p+2}\) or \(\beta'(2p+1)/\pi^{2p+1}\), together with explicit rational linear combinations of \(1\), \(\ln 2\), and \(\ln \pi\). The paper further observes that suitable \(\mathbb{Q}\)-linear independence statements for the even values \(M_1(2m)\) or \(M_2(2m)\), modulo \(\pi\), would imply irrationality results for \(\zeta(2n+1)/\pi^{2n+1}\) or \(\beta(2n)/\pi^{2n}\) [2601.17092]. ## 4. Truncated Mellin moments in perturbative QCD In QCD, truncated Mellin transforms are usually formulated as truncated Mellin moments of parton distributions and structure functions. For a PDF or structure function \(f(x,Q^2)\), the \(n\)th truncated moment is \[ M_n(x_0,Q^2)\equiv \int_{x_0}^{1} dx\,x^{n-1}f(x,Q^2), \] with \(x_0>0\) chosen as the lowest experimentally accessible Bjorken-\(x\). The principal motivation is to avoid model dependence in the unmeasured small-\(x\) region while retaining exact perturbative evolution [1405.5315]. The central result is that the truncated moment obeys a DGLAP-type equation of the same convolutional form as the original PDF, but with a rescaled splitting kernel: \[ \partial_l\,M_n(x_0,Q^2) = \frac{\alpha_s(Q^2)}{2\pi} \int_{x_0}^{1}\frac{dz}{z}\, P'_n(z)\,M_n(x_0/z,Q^2), \qquad P'_n(z)\equiv z^n P(z). \] In shorthand, \[ \partial_l\,M_n(x_0)=\frac{\alpha_s}{2\pi}\,(P'_n\otimes M_n)(x_0). \] This rescaling extends universally through higher perturbative orders: each splitting function \(P^{(k)}(z)\) is replaced by \(P_n^{\prime\,(k)}(z)=z^nP^{(k)}(z)\), and the same rule applies to Wilson coefficient functions, \(C'_n(z)=z^nC(z)\) [1405.5315]. The same framework admits a matrix generalization in the singlet–gluon sector and polarized analogues with \(\Delta P'_{ij}(x)=x^n\Delta P_{ij}(x)\). If \(F(x,Q^2)=(C\otimes f)(x,Q^2)\), then the truncated moment of the structure function satisfies \[ F^n(x,Q^2)=(C'\otimes f^n)(x,Q^2), \qquad C'(x)=x^n C(x). \] For the polarized structure function \(g_1\), this yields an \(n\)th truncated-moment factorization in which the NLO \(\overline{\rm MS}\) coefficient functions are likewise rescaled [1607.08397]. The truncation can also be doubled. Defining \[ \bar q^{\,n}(x_{\min},x_{\max},Q^2) \equiv \int_{x_{\min}}^{x_{\max}} dx\,x^{n-1}q(x,Q^2), \] one obtains a closed Volterra-type evolution equation with the same rescaled kernel \(P'(n,z)=z^nP(z)\). By setting \((x_{\min},x_{\max})=(x_0,1)\) or \((0,1)\), the single-truncated or full-moment equations are recovered [1106.3753]. ## 5. Structural identities and spin-dependent relations The truncated-moment formalism has a substantial internal algebra. One useful construction is the Mellin transform of a truncated moment, \[ M^{\,s,n}(Q^2)\equiv \int_0^1 dx\,x^{s-1}\,\bar q^{\,n}(x,Q^2), \] which satisfies \[ M^{\,s,n}=\frac{1}{s}\,\bar q^{\,s+n}. \] Conversely, \[ \bar q^{\,s}(Q^2) = (s-n)\int_0^1 dx\,x^{s-n-1}\,\bar q^{\,n}(x,Q^2). \] These formulas explicitly link truncated and untruncated moments and show that a Mellin transform in the truncation variable reproduces a full moment of shifted order [1106.3753]. A major application concerns the spin structure function \(g_2\). Starting from the Wandzura–Wilczek relation \[ g_{2}^{WW}(x,Q^2) = -\,g_{1}(x,Q^2)+\int_x^1 \frac{dy}{y}\,g_1(y,Q^2), \] one derives the generalized truncated-moment relation \[ \bar g_{2}^{\,n}(x_0,Q^2) = \frac{1-n}{n}\,\bar g_{1}^{\,n}(x_0,Q^2) - \frac{x_0^{\,n}}{n}\,\bar g_{1}^{\,0}(x_0,Q^2). \] For \(x_0=0\), this reduces to the standard relation \(\bar g_2^{\,n}=(1-n)\bar g_1^{\,n}/n\). For \(n=1\) and finite \(x_0\), \[ \bar g_2^{\,1}(x_0)=-x_0\,\bar g_1^{\,0}(x_0), \] and taking \(x_0\to 0\) reproduces the Burkhardt–Cottingham sum rule \[ \int_0^1 dx\,g_2^{WW}(x)=0. \] The same framework yields a partial-interval Wandzura–Wilczek sum rule for the twist-2 contribution: \[ \int_{x_1}^{x_2} dx\,g_2^{WW}(x) = (x_2-x_1)\int_{x_2}^{1}\frac{dx}{x}g_1(x) - x_1\int_{x_1}^{x_2}\frac{dx}{x}g_1(x). \] Moreover, the leading-twist part of \(g_2\) obeys a standard DGLAP-type evolution equation with the same splitting function as \(g_1\): \[ \frac{d\,g_2^{WW}(x,Q^2)}{d\ln Q^2} = \frac{\alpha_s(Q^2)}{2\pi} \int_x^1 \frac{dz}{z}\, P\!\bigl(\tfrac{x}{z}\bigr)\, g_2^{WW}(z,Q^2). \] This makes the twist-2 evolution of \(g_2\) computationally compatible with ordinary PDF-evolution machinery [1106.3753]. ## 6. Phenomenology in deep-inelastic scattering The practical appeal of truncated Mellin moments is that they are defined directly on the measured \(x\)-range. By choosing \(x_0\) at the experimental lower limit, all integrals are data-driven and no modeling below \(x_0\) is required. This is particularly useful for partial sum rules, for the nonsinglet combination \(\Gamma_1^{p-n}(x_0,Q^2)=\int_{x_0}^1 dx\,[g_1^p-g_1^n](x,Q^2)\), and for reconstruction of parton densities through \[ f(x,Q^2)=-x^{1-n}\,\frac{\partial M_n(x,Q^2)}{\partial x}. \] The method applies to both unpolarized and polarized structure functions and keeps the evolution equations identical in structure to standard DGLAP after the simple kernel rescaling \(P(z)\to z^nP(z)\) [1405.5315]. Numerically, the truncated-moment approach is smooth in the cutoff variable. For typical polarized inputs at \(Q^2=10\,\mathrm{GeV}^2\), cutting at \(x_0=0.01\) retains about \(90\)–\(95\%\) of the full first moment, while at \(x_0=0.001\) one is essentially within \(98\)–\(99\%\). The \(Q^2\) evolution of a fixed-\(x_0\) truncated moment is again DGLAP-like; in the nonsinglet channel, \(\Gamma_1(x_0,Q^2)\) typically increases slowly with \(\ln Q^2\), and the size of the NLO correction is at the level of \(5\)–\(10\%\) for \(Q^2\) between \(1\) and \(100\,\mathrm{GeV}^2\). Higher truncated moments \((n=2,3,4)\) are even less sensitive to the small-\(x\) region [1607.08397]. Using two polarized input scenarios—Input I with moderate small-\(x\) behaviour and Input II with steeper \(\sim x^{-0.4}\)—the truncated-moment analysis was compared directly with HERMES and COMPASS measurements of partial first moments: | Channel | Kinematics | experiment / theory (I) / theory (II) | |---|---|---| | proton | HERMES: \(Q^2=5\,\mathrm{GeV}^2,\ 0.021