Computing differential Galois groups of second-order linear q-difference equations

Computing the N-th term of a q-holonomic sequence

“…Improving algorithms for polynomial and rational solutions of such equations is important for finding q-hypergeometric solutions [4], for computing q-difference Galois groups [17,84], and for performing q-creative telescoping [51,97,98].…”

Mahler discrete residues and summability for rational functions

“…Continuous residues are fundamental and crucial tools in complex analysis, and have extensive and compelling applications in combinatorics [FS09]. In the last decade, a theory of discrete and q-discrete residues was proposed in [CS12] for the study of telescoping problems for bivariate rational functions, and subsequently found applications in the computation of differential Galois groups of second-order linear difference [Arr17] and q-difference equations [AZ22]. More recently, the authors of [Car21,CD75] developed a theory of residues for skew rational functions, which has important applications in duals of linearized Reed-Solomon codes [CD75].…”

“…The discrete and q-discrete residues developed in [CS12] comprise complete obstructions to the summability problem of deciding whether f (x) = g(x+1)−g(x) for some g(x) ∈ K(x) and the q-summability problem of deciding whether f (x) = g(qx)−g(x) for some g(x) ∈ K(x) and q ∈ K neither zero nor a root of unity, respectively. It is precisely this theoretical property of (q-)discrete residues what enables their applications to the telescoping problems considered in [CS12] and their indispensable role in the development of the algorithms in [Arr17,AZ22]. We envision analogous applications of Mahler discrete residues to telescoping problems and in the development of algorithms to compute differential Galois groups for Mahler difference equations.…”

Mahler Discrete Residues and Summability for Rational Functions

“…Continuous residues are fundamental and crucial tools in complex analysis, and have extensive and compelling applications in combinatorics [16]. In the last decade, a theory of (q-)discrete residues was proposed in [14] for the study of telescoping problems, which has found essential applications in several other closely related problems (see [3,4,11,19] for some examples). A theory of residues for skew rational functions was developed in [9], and then extended to Ore polynomials and applied to linearized Reed-Solomon codes in [10].…”

“…[5, §1 and The discrete and q-discrete residues developed in [14] comprise complete obstructions to the summability problem of deciding whether f (x) = д(x + 1) − д(x) for some д(x) ∈ K(x) and the q-summability problem of deciding whether f (x) = д(qx) − д(x) for some д(x) ∈ K(x) and q ∈ K neither zero nor a root of unity, respectively. This theoretical property of (q-)discrete residues is precisely what enables their applications to the telescoping problems considered in [14] and their indispensable role in the development of the algorithms in [3,4]. We envision analogous applications of Mahler discrete residues to telescoping problems and in the development of algorithms to compute (differential) Galois groups for Mahler difference equations.…”