Truncated Mellin Transforms

• Truncated Mellin transforms are variants where the integration domain or series is curtailed, affecting convergence and analytic continuation properties.
• They enable closed-form evaluations in special-function settings via terminating hypergeometric series and facilitate experimentally driven analyses in perturbative QCD.
• This approach underpins rescaled DGLAP evolution equations and bridges mathematical techniques with practical applications in deep inelastic scattering and fractal dynamics.

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 (Qureshi et al., 2019, Kotlorz et al., 2014, Waffo, 23 Jan 2026).

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 $(0,1)$0-range

In QCD notation, the full Mellin moment of a parton density or structure function is

$(0,1)$1

while the truncated counterpart is

$(0,1)$2

When $(0,1)$3, 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 $(0,1)$4 and $(0,1)$5 are treated as supported on $(0,1)$6, and their Mellin transforms converge absolutely for $(0,1)$7 (Kotlorz et al., 2014, Kotlorz et al., 2016, Waffo, 23 Jan 2026).

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 $(0,1)$8 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 (Qureshi et al., 2019).

The starting point is the one-sided Mellin transform with the convergence conditions $(0,1)$9 and, in the exponential case, $x\in[0,1]$0. For the truncated Goursat function, term-by-term integration yields the fundamental identity

$x\in[0,1]$1

The transform problem is thereby reduced to a terminating $x\in[0,1]$2 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 $x\in[0,1]$3, 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: $x\in[0,1]$4, $x\in[0,1]$5, 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 $x\in[0,1]$6 (Qureshi et al., 2019).

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 $x\in[0,1]$7. The transforms

$x\in[0,1]$8

are studied as Mellin transforms of functions supported on $x\in[0,1]$9, with absolute convergence for $[x_0,1]$0 (Waffo, 23 Jan 2026).

Using Taylor expansions on $[x_0,1]$1, one obtains Beta-function representations and then Gamma-series expansions. A convenient form is

$[x_0,1]$2

and

$[x_0,1]$3

valid for $[x_0,1]$4. For even integers $[x_0,1]$5, these specialize to rational binomial-coefficient expansions.

At odd integers $[x_0,1]$6, the substitution $[x_0,1]$7 converts the Mellin integrals into hyperbolic integrals with a $[x_0,1]$8 factor. After binomial expansion and use of the Frullani integral,

$[x_0,1]$9

the transforms are expressed as finite rational combinations of derivatives of the Dirichlet eta and beta functions: $[x_{\min},x_{\max}]$0

Even-integer values require a different method because the relevant improper integrals cannot be dispatched by parity. The contour-integral framework introduces

$[x_{\min},x_{\max}]$1

shows that the vertical sides of a rectangular contour vanish, and evaluates the boundary integral by residues at the poles

$[x_{\min},x_{\max}]$2

A bridge identity then relates the original Mellin-type integrals with $[x_{\min},x_{\max}]$3 factors to these logarithmic hyperbolic integrals: $[x_{\min},x_{\max}]$4 The resulting closed forms involve $[x_{\min},x_{\max}]$5 or $[x_{\min},x_{\max}]$6, together with explicit rational linear combinations of $[x_{\min},x_{\max}]$7, $[x_{\min},x_{\max}]$8, and $[x_{\min},x_{\max}]$9. The paper further observes that suitable $x$0-linear independence statements for the even values $x$1 or $x$2, modulo $x$3, would imply irrationality results for $x$4 or $x$5 (Waffo, 23 Jan 2026).

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 $x$6, the $x$7th truncated moment is

$x$8

with $x$9 chosen as the lowest experimentally accessible Bjorken-$M\{f(t);s\}=\int_0^\infty t^{s-1}f(t)\,dt$0. The principal motivation is to avoid model dependence in the unmeasured small-$M\{f(t);s\}=\int_0^\infty t^{s-1}f(t)\,dt$1 region while retaining exact perturbative evolution (Kotlorz et al., 2014).

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: $M\{f(t);s\}=\int_0^\infty t^{s-1}f(t)\,dt$2 In shorthand,

$M\{f(t);s\}=\int_0^\infty t^{s-1}f(t)\,dt$3

This rescaling extends universally through higher perturbative orders: each splitting function $M\{f(t);s\}=\int_0^\infty t^{s-1}f(t)\,dt$4 is replaced by $M\{f(t);s\}=\int_0^\infty t^{s-1}f(t)\,dt$5, and the same rule applies to Wilson coefficient functions, $M\{f(t);s\}=\int_0^\infty t^{s-1}f(t)\,dt$6 (Kotlorz et al., 2014).

The same framework admits a matrix generalization in the singlet–gluon sector and polarized analogues with $M\{f(t);s\}=\int_0^\infty t^{s-1}f(t)\,dt$7. If $M\{f(t);s\}=\int_0^\infty t^{s-1}f(t)\,dt$8, then the truncated moment of the structure function satisfies

$M\{f(t);s\}=\int_0^\infty t^{s-1}f(t)\,dt$9

For the polarized structure function $(0,1)$0, this yields an $(0,1)$1th truncated-moment factorization in which the NLO $(0,1)$2 coefficient functions are likewise rescaled (Kotlorz et al., 2016).

The truncation can also be doubled. Defining

$(0,1)$3

one obtains a closed Volterra-type evolution equation with the same rescaled kernel $(0,1)$4. By setting $(0,1)$5 or $(0,1)$6, the single-truncated or full-moment equations are recovered (Kotlorz et al., 2011).

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,

$(0,1)$7

which satisfies

$(0,1)$8

Conversely,

$(0,1)$9

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 (Kotlorz et al., 2011).

A major application concerns the spin structure function $\int_0^1 x^{s-1}f(x)\,dx$0. Starting from the Wandzura–Wilczek relation

$\int_0^1 x^{s-1}f(x)\,dx$1

one derives the generalized truncated-moment relation

$\int_0^1 x^{s-1}f(x)\,dx$2

For $\int_0^1 x^{s-1}f(x)\,dx$3, this reduces to the standard relation $\int_0^1 x^{s-1}f(x)\,dx$4. For $\int_0^1 x^{s-1}f(x)\,dx$5 and finite $\int_0^1 x^{s-1}f(x)\,dx$6,

$\int_0^1 x^{s-1}f(x)\,dx$7

and taking $\int_0^1 x^{s-1}f(x)\,dx$8 reproduces the Burkhardt–Cottingham sum rule

$\int_0^1 x^{s-1}f(x)\,dx$9

The same framework yields a partial-interval Wandzura–Wilczek sum rule for the twist-2 contribution: $M_n(x_0,Q^2)=\int_{x_0}^1 x^{n-1}f(x,Q^2)\,dx$0 Moreover, the leading-twist part of $M_n(x_0,Q^2)=\int_{x_0}^1 x^{n-1}f(x,Q^2)\,dx$1 obeys a standard DGLAP-type evolution equation with the same splitting function as $M_n(x_0,Q^2)=\int_{x_0}^1 x^{n-1}f(x,Q^2)\,dx$2: $M_n(x_0,Q^2)=\int_{x_0}^1 x^{n-1}f(x,Q^2)\,dx$3 This makes the twist-2 evolution of $M_n(x_0,Q^2)=\int_{x_0}^1 x^{n-1}f(x,Q^2)\,dx$4 computationally compatible with ordinary PDF-evolution machinery (Kotlorz et al., 2011).

6. Phenomenology in deep-inelastic scattering

The practical appeal of truncated Mellin moments is that they are defined directly on the measured $M_n(x_0,Q^2)=\int_{x_0}^1 x^{n-1}f(x,Q^2)\,dx$5-range. By choosing $M_n(x_0,Q^2)=\int_{x_0}^1 x^{n-1}f(x,Q^2)\,dx$6 at the experimental lower limit, all integrals are data-driven and no modeling below $M_n(x_0,Q^2)=\int_{x_0}^1 x^{n-1}f(x,Q^2)\,dx$7 is required. This is particularly useful for partial sum rules, for the nonsinglet combination $M_n(x_0,Q^2)=\int_{x_0}^1 x^{n-1}f(x,Q^2)\,dx$8, and for reconstruction of parton densities through

$M_n(x_0,Q^2)=\int_{x_0}^1 x^{n-1}f(x,Q^2)\,dx$9

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 $(0,1)$00 (Kotlorz et al., 2014).

Numerically, the truncated-moment approach is smooth in the cutoff variable. For typical polarized inputs at $(0,1)$01, cutting at $(0,1)$02 retains about $(0,1)$03–$(0,1)$04 of the full first moment, while at $(0,1)$05 one is essentially within $(0,1)$06–$(0,1)$07. The $(0,1)$08 evolution of a fixed-$(0,1)$09 truncated moment is again DGLAP-like; in the nonsinglet channel, $(0,1)$10 typically increases slowly with $(0,1)$11, and the size of the NLO correction is at the level of $(0,1)$12–$(0,1)$13 for $(0,1)$14 between $(0,1)$15 and $(0,1)$16. Higher truncated moments $(0,1)$17 are even less sensitive to the small-$(0,1)$18 region (Kotlorz et al., 2016).

Using two polarized input scenarios—Input I with moderate small-$(0,1)$19 behaviour and Input II with steeper $(0,1)$20—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: $(0,1)$21	$(0,1)$22 / $(0,1)$23 / $(0,1)$24
deuteron	HERMES: $(0,1)$25	$(0,1)$26 / $(0,1)$27 / $(0,1)$28
nonsinglet $(0,1)$29	HERMES: $(0,1)$30	$(0,1)$31 / $(0,1)$32 / $(0,1)$33
proton	COMPASS: $(0,1)$34	$(0,1)$35 / $(0,1)$36 / $(0,1)$37
isoscalar $(0,1)$38	COMPASS: $(0,1)$39	$(0,1)$40 / $(0,1)$41 / $(0,1)$42
nonsinglet $(0,1)$43	COMPASS: $(0,1)$44	$(0,1)$45 / $(0,1)$46 / $(0,1)$47

Agreement is typically at the level of a few percent in the proton channel once a reasonable small-$(0,1)$48 input is chosen. This makes truncated moments especially suitable for direct confrontation with partial-moment data, without extrapolating $(0,1)$49 below the measured domain (Kotlorz et al., 2016).

7. Other analytic appearances: harmonic sawtooth maps and zeta-type continuation

Truncated Mellin-transform ideas also appear in a more geometric and dynamical setting. For the harmonic sawtooth map $(0,1)$50 of the unit interval onto itself, the appropriately scaled Mellin transform of $(0,1)$51 is described as an analytic continuation of the Riemann zeta function $(0,1)$52 valid for all $(0,1)$53 not an integer. The inverse scaling function has a series expansion whose coefficients are enumerated by the Large Schröder numbers, and a finite-sum approximation is accompanied by an associated function that solves a reflection formula (Crowley, 2012).

The same work connects Mellin analysis with fractal-string data. The geometric counting function of the fractal string associated to the lengths of the harmonic sawtooth map components coincides with the counting function for the number of Pythagorean triangles of the form $(0,1)$54. The volume of the inner tubular neighborhood of the boundary of the map with radius $(0,1)$55 is stated to have a particularly simple closed form; the Minkowski content is $(0,1)$56 and the Minkowski dimension is $(0,1)$57, and thus not invertible. In the finite reflection problem, the reflection function is singular at $(0,1)$58, and the residue at this point changes sign from negative to positive between $(0,1)$59 and $(0,1)$60 (Crowley, 2012).

This usage does not supply the explicit truncated Mellin formulas in the available summary, but it shows that truncation-based Mellin methods are not confined to hypergeometric identities or parton phenomenology. They also arise in analytic continuations tied to dynamical systems, zeta functions, and fractal geometry.