Transcendental Liouvillian extensions are differential fields, in which one can model poly-logarithmic, hyperexponential, and trigonometric functions, logarithmic integrals, and their (nested) rational expressions. For such an extension $(F, \, ^\prime)$ with the subfield $C$ of constants, we construct a complementary subspace $W$ for the $C$-subspace of derivatives in $F$, and develop an algorithm that, for every $f \in F$, computes a pair $(g,r) \in F \times W$ such that $f = g^\prime + r$. Moreover, $f$ is a derivative in $F$ if and only if $r=0$. The algorithm enables us to determine elementary integrability over $F$ by computing parametric logarithmic parts, and leads to a reduction-based approach to constructing telescopers for functions that can be represented by elements in $F$.

Consider formal power series $f_1,\ldots, f_k\in\mathbb{Q}[[z]]$ that are defined as the solutions of a system of polynomial differential equations together with a sufficient number of initial conditions. Given $P\in \mathbb{Q}[F_1,\ldots,F_k]$, several algorithms have been proposed in order to test whether $P(f_1,\ldots,f_k)=0$. In this paper, we present such an algorithm for the case where $f_1,\ldots,f_k$ are so-called transseries instead of power series.

This paper studies the integration problem in differential fields that may involve quantities reminiscent of the Weierstrass ℘ function, which are defined by a first-order nonlinear differential equation. We extend the classical notion of special polynomials to elements of Weierstrass-like extensions and present algorithms for reduction in such extensions. As an application of these results, we derive some new formulae for integrals of powers of ℘.

Continuing previous work, this paper focuses on the summability problem of multivariate rational functions in the mixed case in which both shift and q-shift operators can appear. Our summability criteria rely on three ingredients including orbital decompositions, Sato's isotropy groups, and difference transformations. This work settles the rational case of the long-term project aimed at developing algorithms for symbolic summation of multivariate functions.

In this paper, we study patterns on permutations of multisets whose multivariate distribution generating functions are symmetric. We interpret this phenomenon through the lens of group actions and define such a pattern as stable. Although various stability results are already implicit in existing enumerative work, we explicitly summarize them here and provide bijective proofs. These bijections offer new combinatorial insight into the symmetry of the generating functions. We also establish instability results. In particular, we provide a complete characterization of stable classical patterns, showing that the only such patterns are those of length one or two. For consecutive patterns, we reprove the stability of all monotone patterns and also identify a large class of unstable patterns. We conjecture that monotone patterns are the only stable consecutive patterns. As an application, we use stability to derive recurrence relations for the ascent distribution over permutations of restricted multisets, yielding a generalization of Eulerian numbers.

Wilf-Zeilberger pairs are fundamental in the algorithmic theory of Wilf and Zeilberger for computer-generated proofs of combinatorial identities. Wilf-Zeilberger forms are their high-dimensional generalizations, which can be used for proving and discovering convergence acceleration formulas. This paper presents a structural description of all possible rational such forms, which can be viewed as an additive analog of the classical Ore-Sato theorem. Based on this analog, we show a structural decomposition of so-called multivariate hyperarithmetic terms, which extend multivariate hypergeometric terms to the additive setting.

Automating formal proofs of combinatorial identities is challenging for LLM-based provers, as long-horizon proof planning is required and unconstrained search quickly explodes. Symbolic methods such as the Wilf--Zeilberger (WZ) method can achieve a mechanized proof of combinatorial identities by constructing special auxiliary functions and demonstrating that they satisfy specific recurrence relations. We propose WZ-LLM, a neuro-symbolic framework that turns WZ proof plans into executable proof sketches in Lean~4 and uses an LLM-based prover to discharge the resulting machine-checkable subgoals. We also train a dedicated WZ-Prover via a Lean-kernel-verified bootstrapping loop with expert-verified iteration, followed by DAPO-based refinement. Experiments show that WZ-LLM achieves a 34\% proof success rate on LCI-Test (100 classical combinatorial identities), outperforming strong baselines such as DeepSeek-V3 and Goedel-Prover-V2; moreover, on LCI-Test it proves 5 identities on which the symbolic-only baseline fails. WZ-LLM also improves performance on CombiBench and PutnamBench-Comb, suggesting the effectiveness of coupling symbolic proof sketches with learned formal reasoning. Experiments show that WZ-LLM achieves a 34\% proof success rate on LCI-Test (100 classic combinatorial identities), outperforming strong baselines such as DeepSeek-V3 and Goedel-Prover-V2, and delivering consistent gains on CombiBench and PutnamBench-Comb. These results indicate that our framework provides two complementary strengths: improved direct proving for identities beyond the scope of WZ, and substantially higher end-to-end success when WZ sketches guide a specialized prover.

We adapt the theory of normal and special polynomials from symbolic integration to the summation setting and then build up a general framework embracing both the usual shift case and the q-shift case. In the context of this general framework, we develop a unified reduction algorithm, and subsequently a creative telescoping algorithm, applicable to both hypergeometric terms and their q-analogues. Our algorithms allow us to split up the usual shift case and the q-shift case only when it is really necessary, and thus instantly reveal the intrinsic differences between these two cases. Computational experiments are also provided.

In this paper, we propose a general framework that extends the theory of permutation patterns to higher dimensions and unifies several combinatorial objects studied in the literature. Our approach involves introducing the concept of a "level" for an element in a multi-dimensional permutation, which can be defined in multiple ways. We consider two natural definitions of a level, each establishing connections to other combinatorial sequences found in the Online Encyclopedia of Integer Sequences (OEIS). Our framework allows us to offer combinatorial interpretations for various sequences found in the OEIS, many of which previously lacked such interpretations. As a notable example, we introduce an elegant combinatorial interpretation for the Springer numbers: they count weakly increasing 3-dimensional permutations under the definition of levels determined by maximal entries.

Symbolic summation as an active research topic of symbolic computation provides efficient algorithmic tools for evaluating and simplifying different types of sums arising from mathematics, computer science, physics and other areas. Most of existing algorithms in symbolic summation are mainly applicable to the problem with univariate inputs. A long-term project in symbolic computation is to develop theories, algorithms and software for the symbolic summation of multivariate functions. This paper will give complete solutions to two challenging problems in symbolic summation of multivariate rational functions, namely the rational summability problem and the existence problem of telescopers for multivariate rational functions. Our approach is based on the structure of Sato’s isotropy groups of polynomials, which enables us to reduce the problems to testing the shift equivalence of polynomials. Our results provide a complete solution to the discrete analogue of Picard’s problem on differential forms and can be used to detect the applicability of the Wilf-Zeilberger method to multivariate rational functions.

These notes on creative telescoping are based on a series of lectures at the Institut Henri Poincare in November and December 2023.

The Shift Equivalence Testing (SET) of polynomials is deciding whether two polynomials $p(x_1, \ldots, x_m)$ and $q(x_1, \ldots, x_m)$ satisfy the relation $p(x_1 + a_1, \ldots, x_m + a_m) = q(x_1, \ldots, x_m)$ for some $a_1, \ldots, a_m$ in the coefficient field. The SET problem is one of basic computational problems in computer algebra and algebraic complexity theory, which was reduced by Dvir, Oliveira and Shpilka in 2014 to the Polynomial Identity Testing (PIT) problem. This paper presents a general scheme for designing algorithms to solve the SET problem which includes Dvir-Oliveira-Shpilka's algorithm as a special case. With the algorithms for the SET problem over integers, we give complete solutions to two challenging problems in symbolic summation of multivariate rational functions, namely the rational summability problem and the existence problem of telescopers for multivariate rational functions. Our approach is based on the structure of isotropy groups of polynomials introduced by Sato in 1960s. Our results can be used to detect the applicability of the Wilf-Zeilberger method to multivariate rational functions.

We propose a way to split a given bivariate P-recursive sequence into a summable part and a non-summable part in such a way that the non-summable part is minimal in some sense. This decomposition gives rise to a new reduction-based creative telescoping algorithm based on the concept of integral bases.

A complete reduction on a difference field is a linear operator that enables one to decompose an element of the field as the sum of a summable part and a remainder such that the given element is summable if and only if the remainder is equal to zero. In this paper, we present a complete reduction in a tower of Σ^*-extensions that turns to a new efficient framework for the parameterized telescoping problem. Special instances of such Σ^*-extensions cover iterative sums such as the harmonic numbers and generalized versions that arise, e.g., in combinatorics, computer science or particle physics. Moreover, we illustrate how these new ideas can be used to reduce the depth of the given sum and provide structural theorems that connect complete reductions to Karr's Fundamental Theorem of symbolic summation.

D-finite power series appear ubiquitously in combinatorics, number theory, and mathematical physics. They satisfy systems of linear partial differential equations whose solution spaces are finite-dimensional, which makes them enjoy a lot of nice properties. After attempts by others in the 1980s, Lipshitz was the first to prove that the class they form in the multivariate case is closed under the operation of diagonal. In particular, an earlier work by Gessel had addressed the D-finiteness of the diagonals of multivariate rational power series. In this paper, we give another proof of Gessel's result that fixes a gap in his original proof, while extending it to the full class of D-finite power series. We also provide a single exponential bound on the degree and order of the defining differential equation satisfied by the diagonal of a D-finite power series in terms of the degree and order of the input differential system.

Elaborating on an approach recently proposed by Mark van Hoeij, we continue to investigate why creative telescoping occasionally fails to find the minimal-order annihilating operator of a given definite sum or integral. We offer an explanation based on the consideration of residues.

Flip graphs were recently introduced in order to discover new matrix multiplication methods for matrix sizes. The technique applies to other tensors as well. In this paper, we explore how it performs for polynomial multiplication.

A complete reduction ϕ for derivatives in a differential field F is a linear idempotent on F over its constant subfield whose kernel is equal to the subspace consisting of derivatives in F. It enables us to decompose an element f as the sum of a derivative and ϕ ( f) such that f is a derivative in F if and only if ϕ ( f) = 0. We outline a complete reduction algorithmically for derivatives in a transcendental Liouvillian extension of the field of rational functions. Typical examples for transcendental Liouvillian extensions are differential fields generated by (poly-)logarithmic functions, hyperexponential functions, as well as the logarithmic integral, the exponential integral and the error function. Such extensions may contain non-elementary and non-D-finite functions.

We propose a summation analog of the paradigm of parallel integration. Using this paradigm, we make some first steps towards an indefinite summation algorithm applicable to summands that rationally depend on the summation index and a P-recursive sequence and its shifts. Under the assumption that the corresponding difference field has no unnatural constants, we are able to compute a bound on the normal part of the denominator of a potential closed form. We can also handle the numerator. Our algorithm is incomplete so far as we cannot predict the special part of the denominator. However, we do have some structural results about special polynomials for the setting under consideration.

We prove a result that can be seen as an analogue of the Pólya–Carlson theorem for multivariate D-finite power series with coefficients in ℚ̅ . In the special case that the coefficients are algebraic integers, our main result says that if F(x_1,… ,x_m)=∑ f(n_1,… ,n_m)x_1^n_1⋯ x_m^n_m is a D-finite power series in m variables with algebraic integer coefficients and if the logarithmic Weil height of f(n_1,… ,n_m) is o(n_1+⋯ +n_m) , then F is a rational function and, up to scalar multiplication, every irreducible factor of the denominator of F has the form 1-ζ x_1^q_1⋯ x_m^q_m where ζ is a root of unity and q_1,… ,q_m are nonnegative integers, not all of which are zero.