{"id":"44435d6b-5a5e-4b9d-bab3-8533c6482084","arxiv_id":"2608.00781","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A full-rank q×p matrix weight on the unit circle with reciprocal-Gamma moments is introduced whose mixed-type Bessel-like orthogonal polynomials are explicitly computed, with normality, Rodrigues, recurrence, and Christoffel factorization results.","lead":"This paper builds a new family of matrix-valued orthogonal polynomials (Bessel-like, mixed type) with explicit formulas for the polynomials, their recurrences, and matrix factorizations. It also shows that a simple rank-one system can converge to a genuinely matrix-valued one after switching from an interval to a circle contour, a structural insight for approximation theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 7's bidiagonal factorization is imported from unpublished [5], so the strongest version of explicit solvability cannot be independently checked from this manuscript.","rationale":"I read the paper in good faith and found no internal inconsistency in the core derivation of the explicit polynomial vectors, the residue-based orthogonality proofs, the weak-normality argument, or the finite pairing block for recurrences. The reader's weakest assumption concerned the restricted regular admissible domain; that is a real limitation, but it is explicitly declared and does not by itself break the stated theorems. My load-bearing concern is different: the complete bidiagonal factorization, which is part of the paper's advertised explicit solvability, rests on the author's own unpublished [5]. The numerical check in the manuscript is too small and too unaccompanied by code to substitute for an independent derivation. If [5] becomes available and its hypotheses cover this setting, the concern dissolves; until then CONDITIONAL is the right verdict. I mark agreement as partial because the reader did identify the Section 7 import as a secondary issue, though their primary weakest assumption was the admissible-domain restriction.","tokens_in":80425,"tokens_out":10081,"duration_ms":115817,"concrete_test":"For the p=3,q=2 example of Section 7.5, extend the check to N=8 and N=11: compute T_N entrywise from Corollary 6.3 using the pairing-block formula; independently construct the Gauss–Borel factorization of the truncated N x N moment matrix and the corresponding column/row Christoffel perturbations directly from the bilinear form, without invoking [5]; then verify the factorization T_N = L_1 L_2 L_3 U_2 U_1 to high precision, and separately verify the tau-determinant formula (93) by solving the row-perturbed orthogonality problem numerically. Ship the scripts and data so the test is reproducible.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is not just the hypergeometric vector formulas of Theorem 3.1; it also includes the recurrence coefficients of Section 6 and the complete bidiagonal Christoffel factorization of Section 7. Sections 3 and 6 are essentially self-contained: the orthogonality proofs, normality arguments, and finite Gamma–Pochhammer pairing block can be checked from the displayed identities. Section 7 is different. Equation (92), T_N = L_1...L_p U_q...U_1, is asserted by invoking 'the general mixed Christoffel–Gauss–Borel factorization of [5]', and the upper-factor formula (93) is likewise imported from [5]. Reference [5] is listed as 'to appear' and the companion reference [6] as 'in preparation'. The numerical check in Section 7.5 covers only a 5x5 truncation for p=3,q=2, with no shipped code or data, so it does not verify the general factorization. Thus the assertion that every balanced near-diagonal Bessel-like problem is explicitly and completely solvable, including its bidiagonal factorization, is conditional on an unavailable external theorem and on its precise hypotheses matching the present admissible regular parameter domain. This does not indicate an internal contradiction in Theorem 3.1, but it makes the strongest form of the central claim unverifiable from this manuscript alone.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines a q×p Bessel-like matrix weight W on the unit circle with entries w_{j,i}(z)=f_0(z; a+κ_i 1_r, b+κ_i 1_q+e_j), whose moments are reciprocal Gamma quotients. For balanced near-diagonal indices it obtains explicit terminating hypergeometric formulas for the A- and B-polynomial vectors (Theorem 3.1), proves weak normality (Proposition 3.5) and characterizes strong normality componentwise, including exceptional degree-loss loci (Propositions 3.8, 3.9). It also gives Kampé de Fériet representations, a Rodrigues-type formula, reductions to the multiple Bessel and Wolfs systems, finite Gamma–Pochhammer formulas for near-diagonal and step-line recurrence coefficients, and a bidiagonal Christoffel factorization of the banded recurrence matrix. Sections 2–6 are largely self-contained and contain detailed residue, partial-fraction, and finite-difference proofs. Section 7, however, relies on the unpublished companion paper [5] for the central factorization identity (92) and the upper-factor formula (93).","tokens_in":80677,"tokens_out":7904,"duration_ms":99064,"significance":"If correct, the paper gives an explicit, parameter-free family of mixed-type multiple orthogonal polynomials for a genuinely matrix-valued weight, recovering the multiple Bessel and Wolfs systems at the two boundary cases. The strengths are substantial: Theorem 3.1 is proved by direct residue computations, Lemma 3.3 is a long telescoping barycentric identity, Proposition 3.5 gives a finite rational characterization of weak normality, Theorem 2.11 provides a quantitative locally uniform confluence proof, and the recurrence coefficients in Section 6 are evaluated by explicit finite Pochhammer sums. The explicit exceptional loci of Proposition 3.9 are a useful, checkable output. The main caveat is that the bidiagonal factorization asserted in Section 7 is imported from [5] ('to appear') and [6] ('in preparation'), so the strongest form of the paper's central claim is not independently verifiable from this manuscript alone.","major_comments":[{"comment":"The factorization T_N = L_1...L_p U_q...U_1 and the upper-factor formula (93) are asserted by invoking 'the general mixed Christoffel–Gauss–Borel factorization of [5]', with [5] listed as 'to appear'. The numerical check in Section 7.5 covers only one 5×5 truncation. Because the abstract and conclusions present the bidiagonal factorization as part of the main result, this is load-bearing. Please either provide a self-contained proof of (92)–(93) adapted to the present hypotheses, or explicitly label Section 7 as conditional and remove the factorization from the abstract/conclusion claims.","section":"Section 7.2, Eqs. (92)–(93)"},{"comment":"The section begins by assuming that the moment matrix and the Christoffel-perturbed moment matrices admit Gauss–Borel factorizations. For the closed column chain, Lemma 7.1 supplies the moment closure, but for the intermediate row chain the perturbed matrices are not shown to lie in the Bessel-like family or to satisfy an explicit nonvanishing determinant condition. Thus the hypotheses of the imported factorization theorem are not verified. Please prove these hypotheses for the Bessel-like weights, or state them as explicit additional assumptions and delimit their scope.","section":"Section 7.2, Gauss–Borel assumption"},{"comment":"The p=3, q=2 numerical check verifies the factorization only for a 5×5 truncation, reports a residual of 1.1×10^-73, and ships no code or data. This is reasonable supporting evidence but cannot validate the general factorization theorem, particularly since (92)–(93) are imported from an unpublished reference. It should be presented as an illustration rather than as independent verification of the general statement.","section":"Section 7.5, numerical example"}],"minor_comments":[{"comment":"The paper clearly states that every formula-based assertion is made on the F-admissible subset and that no analytic continuation is intended. This is a legitimate scope restriction, but it means the explicit formulas exclude integer separations b_j-b_h∈Z and κ_i-κ_h∈Z as well as Gamma-denominator poles. The abstract could say this more prominently, since the word 'explicit' is otherwise easy to overread.","section":"Definition 2.1, admissibility convention"},{"comment":"The confluence theorem is self-contained once the beta-convolution weights are defined, but the claim that the starting system is the Jacobi-like system of [6] cannot be checked because [6] is listed as 'in preparation'. This does not affect the main theorems, but it makes the historical/comparative statement unverifiable.","section":"Section 2.2, Jacobi-like system of [6]"},{"comment":"The multiple Bessel reduction q=1 identifies α_j=b_1+κ_j. The moment formula (59) is consistent with (8), but the notation L_j[z^n] is introduced before the functional is defined; a short pointer to (8) would improve readability.","section":"Section 5, Eq. (59) vs. Eq. (8)"},{"comment":"The displayed definition of 1_d is typeset as [1 1] rather than [1 ... 1]; this is a minor typographical issue. Also, the double use of * for deletion of a component (®c_*i) and for step-line quantities could be flagged in the notation section.","section":"Notation, Section 1.3"}],"recommendation":"major_revision","confidential_remarks":"The central Sections 2–6 are detailed and credible; the main risk is Section 7's dependence on the unpublished [5] and [6]. I would advise the editor that the paper can be accepted if the author either supplies a self-contained proof of the factorization or reframes the contribution around the polynomial-vector, normality, Rodriguez, and recurrence results, clearly marking the factorization as conditional. There is no apparent circularity in the main derivations, and the paper is careful about its admissibility limitations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nHere's the short version: this is a real piece of constructive work. The q×p matrix weight with independent non-integral column shifts has maximal rank away from finitely many points; the balanced near-diagonal A- and B-polynomial vectors are given in terminating hypergeometric form, with orthogonality and weak normality proved by direct residue arguments. I checked the logic in Sections 2–3 and did not find an internal contradiction. The proof of Theorem 3.1 is self-contained: the moments are computed from the declared weight, the polynomial coefficients are residues, and the orthogonality is a residue cancellation. The claim that the weight is not rank-one, and the Jacobi confluence in Theorem 2.11, are both clearly argued.\n\nThe soft spots are real but not fatal. Section 7's bidiagonal factorization (equation 92) imports the central Christoffel–Gauss–Borel factorization from [5], which is \"to appear\", and [6] is \"in preparation\". So the strongest \"explicit complete solvability\" claim cannot be independently checked from this manuscript. The numerical check is one 5×5 truncation, no code or data shipped. Also, the formulas are proven only on the admissible regular domain, with no analytic continuation; the author says this explicitly, so it is not a hidden flaw, but it does limit the footprint.\n\nIf I were the editor, I would send it to a serious referee. The core of the paper deserves scrutiny, and the referee should ask the author to either include the needed statement from [5] or clearly mark the factorization results as conditional on it. That is a normal request, not a rejection.\n\nIt is worth citing if you work on explicit systems of multiple orthogonal polynomials; I would not cite Section 7 until the companion appears.","headline":"A serious, largely self-contained construction of explicit mixed-type Bessel-like multiple orthogonal polynomials; the core orthogonality and normality proofs hold up, but the bidiagonal factorization results depend on unpublished companion work and should be conditioned on it.","tokens_in":81245,"tokens_out":1713,"would_cite":true,"duration_ms":22108,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["33C45","42C05","33C20","15A23","47B36"],"pacs":[],"model":"deepseek-v4-flash","headline":"A matrix of Bessel-like weights on the unit circle makes every balanced near-diagonal mixed-type orthogonality problem explicitly solvable.","keywords":["mixed-type multiple orthogonal polynomials","Bessel polynomials","matrix weights on the unit circle","reciprocal-Gamma moments","near-diagonal multi-indices","bidiagonal Christoffel factorization","multiple Hahn polynomials","Gamma–Pochhammer pairing"],"falsifier":"Run the Example 3.10 computation in exact arithmetic: for p=q=2, r=1, n=(2,2), m=(3,2), b1=1/10, κ1=1/5, κ2=19/20, a1=3/10, and b2 the real root in (−0.81,−0.80) of 12800 x^3+171520 x^2−1674878 x−1447069, the rational identity of Theorem 3.1 must give top coefficients β_{1,2}≠0 and β_{2,1}=0; any exact rational calculation giving β_{2,1}≠0 would refute the exceptional-equation description of Proposition 3.9. More generally, evaluating the claimed hypergeometric A- and B-vectors at a regular admissible parameter tuple and numerically checking that the moment integrals (22)–(23) vanish would set","tokens_in":80215,"feed_emoji":"🧮","tokens_out":8709,"duration_ms":89496,"temperature":0.7,"pith_summary":"This paper constructs a q×p matrix of weights on the unit circle that behaves like a Bessel weight in mixed-type multiple orthogonality: each entry is a base hypergeometric moment-generating function with shifted Gamma parameters, so its Laurent moments are reciprocal-Gamma quotients. Unlike a separable product weight, this matrix has generic rank min{q,p} away from finitely many points. The paper proves that for balanced near-diagonal index pairs, both the A- and B-polynomial vectors are given explicitly by terminating generalized hypergeometric formulas, that their solution spaces are unique up to a scalar (weak normality), and that strong normality holds except on explicit algebraic loci. It then derives finite Gamma–Pochhammer formulas for recurrence coefficients and a bidiagonal Christoffel factorization of the banded step-line recurrence matrix. If correct, the paper makes every balanced near-diagonal mixed orthogonality problem for this family, including its recurrences and Christoffel factors, explicitly computable.","feed_headline":"Explicit solutions for every balanced near-diagonal Bessel problem","feed_subtitle":"Reciprocal-Gamma moments yield explicit polynomial vectors, recurrences, and Christoffel factors.","key_machinery":"The engine of the paper is the reciprocal-Gamma moment pairing (21), with M_{(u,j),(v,i)} = Γ(u+v+1+κ_i+a)/Γ(u+v+1+κ_i+b+e_j), together with the matrix weight W whose entries are the hypergeometric generating function f_0 with shifted parameters. The proofs run through partial-fraction decompositions of the rational function R(t) = -∏(κ_i+1-t)_{n_i}/∏(t+b_j)_{m_j} (Lemma 3.2), the finite reconstruction of a simple fraction (Lemma 3.3), and a residue/interpolation characterization that forces any solution to coincide with the explicit vector (Proposition 3.5). Rank maximality of the weight is established by a Cauchy determinant of the residues (Proposition 2.8). The later sections are carried","core_discovery":"The central claim is that the q×p Bessel-like matrix weight W with entries w_{j,i}(z)=f_0(z; a+κ_i 1_r, b+κ_i 1_q+e_j), whose moments are reciprocal-Gamma quotients (8), supports an explicitly solvable mixed-type orthogonality theory. Theorem 3.1 gives terminating hypergeometric formulas for the A- and B-polynomial vectors satisfying the orthogonality conditions (22)–(23) for every balanced near-diagonal index pair; Proposition 3.5 proves the solution spaces are one-dimensional, and Propositions 3.6 and 3.9 characterize strong normality componentwise, including the exceptional loci where a B-component loses its maximal degree. The B-components also admit finite Kampé de Fériet representation","pith_inferences":["Editorial inference: the same confluence route — rank-one interval weights to a Markov–Stieltjes contour representative that survives while the interval measure does not — may produce full-rank matrix weights for other classical hypergeometric families with non-integer exponents, not just Bessel.","Editorial inference: because the exceptional loci F_j=0 are explicit algebraic equations, one could locate parameter regions where B-components systematically lose degree and then test whether those regions correspond to zeros of the associated multiple Hahn polynomials or to changes in the banded recurrence matrix.","Editorial inference: the explicit Gamma–Pochhammer form of the recurrence coefficients invites an asymptotic and spectral analysis of the banded matrix, potentially yielding large-degree limits for the mixed-type polynomials, although asymptotics are not developed in the paper."],"forward_implications":["Every balanced near-diagonal mixed orthogonality problem for this Bessel-like matrix weight has explicit polynomial vectors, so the A- and B-vectors on the step-line and their duals are obtained by evaluating terminating hypergeometric sums rather than solving linear systems.","Weak normality is guaranteed on the regular admissible parameter domain; strong normality is guaranteed except on the finite union of explicit algebraic hypersurfaces F_j=0 where a B-component drops one degree.","The step-line recurrence matrix and the local near-diagonal recurrences of length p+q+1 have coefficients given by finite Pochhammer sums, so the banded recurrence is computable entry by entry.","The banded recurrence matrix admits a bidiagonal Christoffel factorization; for q=1 the factorization is complete in terms of Gamma–Vandermonde determinants, giving positive, O(n^{-1}) bidiagonal coefficients in the ordered chamber and hence a bounded banded operator.","In the one-row reduction, the type-II multiple Bessel polynomial is expressed as a terminating generalized hypergeometric polynomial whose unit-difference parameters are the roots of a reflected type-II multiple Hahn polynomial, connecting the Bessel and Hahn worlds."],"supporting_citations":[{"why":"Supplies the Gauss–Borel factorization and mixed-type orthogonality framework that the step-line recurrence and dual biorthogonality rely on.","marker":"[1]"},{"why":"Supplies the multiple Bessel system recovered in the q=1 reduction, including its moments and Rodrigues-type identities.","marker":"[2]"},{"why":"Supplies the type-II multiple Hahn polynomials that appear in the one-row reduction's characteristic polynomial.","marker":"[3]"},{"why":"Supplies the mixed Christoffel–Gauss–Borel bidiagonal factorization scheme applied to the banded recurrence matrix.","marker":"[5]"},{"why":"Supplies the rank-one Jacobi-like and Laguerre-like mixed systems whose Markov–Stieltjes representative converges to the Bessel-like weight.","marker":"[6]"},{"why":"Supplies the Exton-type cut-contour evaluation used to identify the reciprocal-Gamma moments in the q=1 case.","marker":"[10]"},{"why":"Supplies the Krall–Frink unit-circle realization of ordinary Bessel orthogonality that motivates the circular contour representative.","marker":"[14]"},{"why":"Supplies the general Christoffel perturbation framework and tau-determinant formulas used for the non-closed row chain.","marker":"[15]"},{"why":"Supplies the Bessel-like circle weights recovered in the p=1 reduction and the base hypergeometric moment-generating function f_0.","marker":"[25]"}],"fun_headline_variants":["Explicit closed forms for balanced Bessel-like orthogonal polynomials","Mixed-type Bessel polynomials: every balanced index explicit","Closed forms for all balanced Bessel-like polynomial vectors","Explicit recurrences and Christoffel factors for Bessel-like systems","Bessel-like polynomials: all balanced near-diagonal solved"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"Everything rests on the parameters being regular and admissible: no two b-parameters and no two κ-parameters may differ by an integer, and every Gamma factor appearing in the displayed formulas must be finite; the paper explicitly says that without analytic continuation every formula-based assertion is restricted to that domain.","fun_headline_variants_meta":{"raw":{"variants":["Explicit closed forms for balanced Bessel-like orthogonal polynomials","Mixed-type Bessel polynomials: every balanced index explicit","Closed forms for all balanced Bessel-like polynomial vectors","Explicit recurrences and Christoffel factors for Bessel-like systems","Bessel-like polynomials: all balanced near-diagonal solved"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001029,"raw_usage":{"total_tokens":4217,"prompt_tokens":830,"completion_tokens":3387,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":574,"completion_tokens_details":{"reasoning_tokens":3305}},"tokens_in":574,"tokens_out":3387,"duration_ms":23174,"temperature":1.0,"reasoning_tokens":3305,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T00:18:13.144542+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the Example 3.10 computation in exact arithmetic: for p=q=2, r=1, n=(2,2), m=(3,2), b1=1/10, κ1=1/5, κ2=19/20, a1=3/10, and b2 the real root in (−0.81,−0.80) of 12800 x^3+171520 x^2−1674878 x−1447069, the rational identity of Theorem 3.1 must give top coefficients β_{1,2}≠0 and β_{2,1}=0; any exact rational calculation giving β_{2,1}≠0 would refute the exceptional-equation description of Proposition 3.9. More generally, evaluating the claimed hypergeometric A- and B-vectors at a regular admissible parameter tuple and numerically checking that the moment integrals (22)–(23) vanish would set","supporting_citations":[{"cited_title":"Arvesú, J","cited_arxiv_id":null,"evidence_quote":"Supplies the type-II multiple Hahn polynomials that appear in the one-row reduction's characteristic polynomial."},{"cited_title":"Jacobi-like and Laguerre-like mixed-type multiple orthogonal polynomials","cited_arxiv_id":null,"evidence_quote":"Supplies the rank-one Jacobi-like and Laguerre-like mixed systems whose Markov–Stieltjes representative converges to the Bessel-like weight."},{"cited_title":"Exton.On the orthogonality of the Bessel polynomials","cited_arxiv_id":null,"evidence_quote":"Supplies the Exton-type cut-contour evaluation used to identify the reciprocal-Gamma moments in the q=1 case."},{"cited_title":"Transactions of the American Mathematical Society65(1949) 100–115.doi: 10.1090/ S0002-9947-1949-0028473-1","cited_arxiv_id":null,"evidence_quote":"Supplies the Krall–Frink unit-circle realization of ordinary Bessel orthogonality that motivates the circular contour representative."},{"cited_title":"Mañas and M","cited_arxiv_id":null,"evidence_quote":"Supplies the general Christoffel perturbation framework and tau-determinant formulas used for the non-closed row chain."}],"review_version":1}