{"id":"ae722cc6-957d-440b-8356-341d3d6a096a","arxiv_id":"2411.16022","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For mixed multiple orthogonal polynomials, the paper derives Christoffel-type formulas for general Geronimus perturbations and proves that the perturbed orthogonality exists exactly when certain tau-determinants do not vanish.","lead":"This mathematics paper extends the classical Geronimus transformation, which modifies a measure by dividing by a polynomial, to systems with several weight functions and matrix-valued polynomial factors. It gives explicit formulas relating the new orthogonal polynomials to the old ones, and a determinant condition that decides when such a transformation preserves orthogonality.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The converse half of the claimed equivalence is undermined by an inconsistent determinant definition: Definition 3.3 defines a 2×2 corner determinant, but the proof requires the full leading principal minors of the matrix I.","rationale":"The reader identified the support condition sigma(R)∩Delta=∅ as the weakest assumption. That condition is indeed necessary for the displayed definition of the perturbed measure in Eq. (2), and it is stated explicitly before the definition; while it restricts the scope, it is a clearly declared hypothesis rather than an internal gap. My concern, by contrast, is an internal inconsistency in the proof of the converse direction of the main equivalence. Definition 3.3 defines tau-tilde_n as a 2×2 determinant built from the first and last rows and columns of the matrix of integrals I, but the proof immediately after needs the full leading principal minors of that matrix to invert the system for the connection coefficients. The Schur complement displayed in the proof is not the ratio of the 2×2 corner determinants defined earlier. This is not merely a typographical nuisance: if the definition is taken literally, tau-tilde_0 vanishes identically, so the proof cannot even begin. If the definition is meant to be the full leading principal minor, that needs to be stated and the nonvanishing of those minors must be derived from the assumed nonvanishing of the tau-determinants. Without this, Theorem 3.5 is not established, and the abstract's equivalence claim is not fully proven. The paper otherwise contains a plausible derivation, and the only-if direction appears sound, so the appropriate verdict remains CONDITIONAL: the manuscript should be revised to correct the determinant definition and complete the converse proof. I disagree with the reader that the support condition is the most load-bearing issue, since that assumption is explicit and standard in Geronimus-perturbation theory; the determinant gap is more central to the paper's main theorem.","tokens_in":40269,"tokens_out":9307,"duration_ms":84518,"concrete_test":"Check the determinant identity used in the proof of Theorem 3.5 by computing it in a concrete case. Take a generic 3×3 matrix I with entries I_{i,l}, i,l=0,1,2, and compare the Schur complement S = I_{2,2} - [I_{2,0}, I_{2,1}] [[I_{0,0}, I_{0,1}],[I_{1,0}, I_{1,1}]]^{-1} [I_{0,2}; I_{1,2}] with the quotient of the corner determinants appearing in Definition 3.3, namely det[[I_{0,0}, I_{0,2}],[I_{2,0}, I_{2,2}]] / det[[I_{0,0}, I_{0,1}],[I_{1,0}, I_{1,1}]]. For a generic matrix these two quantities differ, and for the moment matrix I arising from the Jacobi-Piñeiro example in §2.5 they will differ as well; this would demonstrate that the object used in the proof is not the object defined.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is an equivalence: Geronimus-perturbed orthogonality exists if and only if the tau-determinants do not vanish. The only-if direction is Theorem 2.12 supplemented by Proposition 3.4. The if-direction is Theorem 3.5, whose proof relies on Theorem 3.1 and on a low-degree construction using the tilde-tau determinants introduced in Definition 3.3. There, tau-tilde_n is explicitly defined as the 2×2 determinant with entries I_{0,0}, I_{0,n}, I_{n,0}, I_{n,n}. But immediately after, to determine the connection coefficients Omega_{n,0},...,Omega_{n,n-1} for n<Np-r, the full n×n matrix [I_{i,l}]_{i,l=0}^{n-1} is inverted, and the orthogonality condition at l=n is written as I_{n,n} - [I_{n,0}...I_{n,n-1}][I_{i,l}]_{i,l=0}^{n-1}]^{-1}[I_{0,n};...;I_{n-1,n}] = tau-tilde_n/tau-tilde_{n-1}. The left-hand side is the Schur complement and equals the ratio of the leading principal minors of sizes n+1 and n, not the ratio of the 2×2 corner determinants from Definition 3.3. In particular, for n=0 the displayed definition gives tau-tilde_0 = det[[I_{0,0},I_{0,0}],[I_{0,0},I_{0,0}]] = 0 identically, so the formula tau-tilde_n/tau-tilde_{n-1} is not even meaningful at n=0 as written. Thus the proof that non-vanishing tau_n supplies the vector polynomials required by Theorem 3.1 is incomplete. Because the abstract's main theorem asserts equivalence, this is a load-bearing gap, though it may be repairable by redefining tau-tilde_n to be the full leading principal minor and proving its nonvanishing.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general theory of Geronimus transformations for mixed multiple orthogonal polynomials. For a regular matrix polynomial R(x) satisfying condition (C2) and with no zeros on the support of the measure, the authors define a Geronimus-perturbed matrix of measures by right (and later left) multiplication, with delta masses carrying free parameters. They derive explicit Christoffel-type formulas expressing the perturbed type I and type II polynomials in terms of the original polynomials and certain tau-determinants, and they claim that the existence of the perturbed orthogonality is equivalent to the non-vanishing of these tau-determinants. The paper also computes the effect on Markov-Stieltjes matrix functions and works out the Jacobi-Pineiro example with three weights.","tokens_in":40668,"tokens_out":13319,"duration_ms":126399,"significance":"If correct, the paper would give a useful and fairly general framework for Geronimus perturbations of mixed multiple orthogonal polynomials, extending the authors' earlier Christoffel paper [40] and avoiding restrictive assumptions such as monicity or rank conditions on the leading coefficient. The derivation is constructive and parameter-free with respect to the mass amplitudes, and the only-if direction (Theorem 2.12) is structurally convincing: the residue argument from a vanishing determinant to a zero connection vector is a genuine nontrivial step. The explicit Jacobi-Pineiro example is a valuable concrete illustration. However, the converse half of the central equivalence, and one of the main Christoffel formulas as printed, have load-bearing defects that must be repaired before the claims can be accepted.","major_comments":[{"comment":"The printed definition of tilde-tau_n as the 2x2 corner determinant det[[I_{0,0},I_{0,n}],[I_{n,0},I_{n,n}]] is inconsistent with the computation that immediately follows. The displayed identity I_{n,n} - [I_{n,0} ... I_{n,n-1}] [I_{i,l}]_{0<=i,l<=n-1}^{-1} [I_{0,n} ... I_{n-1,n}]^T = tilde-tau_n / tilde-tau_{n-1} is the Schur complement formula and equals the ratio of the leading principal minors of sizes n+1 and n, not the ratio of 2x2 corner determinants. Moreover, for n=0 the printed definition gives tilde-tau_0 = det[[I_{0,0},I_{0,0}],[I_{0,0},I_{0,0}]] = 0 identically, so the ratio is not even meaningful at n=0. Since Theorem 3.5 and hence the 'if' direction of the abstract equivalence rest on this step, the converse is not proven as written. A redefinition of tilde-tau_n as the full leading principal minor det(I_{i,l})_{0<=i,l<=n} appears to repair the argument, but this must be stated explicitly and the subsequent invertibility assertions adjusted accordingly.","section":"Section 3, Definition 3.3 and following paragraph"},{"comment":"The determinant in Eq. (9) is non-square as printed: it has M rows (the M-1 rows indexed n-M+1 through n-1 plus the row containing the K-terms) but M+1 columns (the M columns D^(1),...,D^(M) plus the K-column). A non-square determinant is undefined, and Eq. (9) is one of the two central Christoffel formulas of the paper. The analogous formula in Theorem 2.21, Eq. (11), appears to have the same problem. The intended square determinant should be stated unambiguously, for example by specifying which column or row is replaced in the underlying M x M linear system.","section":"Theorem 2.13, Eq. (9)"},{"comment":"As printed, the linear system in Proposition 2.7 has incompatible block dimensions. The row block [Omega_{n,n-M} ... Omega_{n,n-1}] is a p x pM block row, since each Omega_{n,r} is p x p by Proposition 2.2, while the matrix with entries D^(i)_r - W^(i)_r is displayed as an M x M scalar matrix. Unless the entries D^(i)_r - W^(i)_r are intended to be p x 1 or p x p blocks, which Definition 2.6 does not state, the product is undefined. Because this system is the origin of the connection coefficients used in Eqs. (8)-(10), the statement needs a consistent block-calibrated formulation.","section":"Proposition 2.7 and Theorem 2.13, Eq. (8)"}],"minor_comments":[{"comment":"The proof defines two new families of matrix polynomials but gives both the name tilde-A: 'tilde-A(x)=R(x)A(x), tilde-A(x)=check-A(x)Omega'. The second symbol should be different, otherwise the subsequent argument is ambiguous.","section":"Theorem 3.1 proof"},{"comment":"There is a missing closing parenthesis in the superscript D^{i)}_{n-M}; it should be D^{(i)}_{n-M}.","section":"Proposition 2.11"},{"comment":"The notation Omega_{n.n-M} uses a period instead of a comma between the indices; it should be Omega_{n,n-M}.","section":"Theorem 2.13 proof"},{"comment":"The index range n in {M-1, M, M-2, ... } should presumably be {M-1, M, M+1, ... }; as written it contains a typographical reversal.","section":"Definition 2.28"},{"comment":"The displayed formula for I_{i,l} contains a redundant d-check-mu inside the integral and reuses the summation index i in the second term; clarifying the indices would improve readability.","section":"Definition 3.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the central idea is promising. I do not see circularity or a fundamental obstruction, and the fixes for the two determinant-related issues appear local. However, the currently printed converse theorem is not proven because of the tilde-tau definitional mismatch, and one of the main formulas is non-square. These are load-bearing and should be corrected before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Manas and Rojas have written a worthwhile follow-up to their Christoffel paper. The general Geronimus setting—non-monic matrix polynomials, arbitrary leading rank, left and right multiplication—is genuinely more general than what was in [17] or [7,8], and the explicit Christoffel-type formulas (8)-(10) and the Jacobi-Piñeiro example are concrete and checkable. The forward direction, Theorem 2.12 and Proposition 3.4, is clear: if the perturbed orthogonality exists, the tau-determinants have to be nonzero. I have no serious quibble with that part. The standing assumption sigma(R) ∩ Δ = ∅ is standard for Geronimus transformations and is stated clearly; it is a structural condition, not a weakness.\n\nThe problem is the converse. Theorem 3.5 says nonvanishing tau_n for all n implies the perturbed orthogonality exists. The proof goes through the auxiliary matrices I_{i,l} and a set of determinants called tau-tilde. Definition 3.3 defines tau-tilde_n as the 2x2 corner determinant with entries I_{0,0}, I_{0,n}, I_{n,0}, I_{n,n}. But a few lines later the proof uses the identity I_{n,n} - [I_{n,0} ... I_{n,n-1}] M_{n-1}^{-1} [I_{0,n};...;I_{n-1,n}] = tau-tilde_n / tau-tilde_{n-1}. That left-hand side is the Schur complement of the leading (n+1)x(n+1) principal submatrix of the I matrix, so it equals the ratio of the leading principal minors of sizes n+1 and n—not the ratio of the 2x2 corner determinants. For n=0 the displayed definition gives tau-tilde_0 = 0 identically, so the ratio isn't even defined. This is not a cosmetic typo; it is the step that connects the nonvanishing of the main tau's to the solvability of the linear system for the connection coefficients.\n\nI think the fix is straightforward: redefine tau-tilde_n to be the n-th leading principal minor of the I matrix, and then show, presumably via Proposition 2.7, that these minors are nonzero when the original tau's are. But as it stands, the if-direction of the main equivalence is not proven. There are also smaller irritants: equation (9) is typeset as a non-square determinant, Proposition 2.11 has a missing parenthesis, and Theorem 3.1 has a duplicate tilde-A. Those are minor.\n\nThe paper deserves a serious referee. The ideas are sound, the framework is natural, and the forward direction plus the explicit formulas are enough to be worth publishing after the converse gap is repaired. I would send it out, but with a clear instruction to the referee to verify the tau-tilde definitions and the Schur complement step.","headline":"A useful extension with a real gap in the converse theorem: the tau-tilde definitions don't match the proof's Schur complement.","tokens_in":41219,"tokens_out":4832,"would_cite":false,"duration_ms":41124,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42C05","33C45","33C47","47B39","47B36"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that, for Geronimus perturbations defined by a regular matrix polynomial with no zeros on the support, the perturbed mixed multiple orthogonality exists if and only if the associated tau-determinants are all nonzero, and…","keywords":["mixed multiple orthogonal polynomials","Geronimus transformations","Christoffel-type formulas","tau-determinants","matrix polynomials","Jordan chains","Markov-Stieltjes matrix functions","Jacobi-Piñeiro polynomials"],"falsifier":"The cleanest test is the scalar case $p=q=1$ with $R(x)=x-a$ for $a$ outside the support: the Geronimus measure is $d\\check{\\mu}=d\\mu/(x-a)+\\xi\\,\\delta(x-a)$. Compute the leading principal minors of the perturbed moment matrix and compare their vanishing with the tau-determinants $\\tau_n$ defined in Definition 2.8 (here $M=1$, so each $\\tau_n$ is the single entry $D_n(a)-W_n(a)$ up to normalization). Any mismatch between a zero $\\tau_n$ and a singular perturbed moment submatrix would disprove the claimed equivalence; the paper predicts exact coincidence.","tokens_in":40038,"feed_emoji":"🧮","tokens_out":11285,"duration_ms":97116,"temperature":0.7,"pith_summary":"The paper proves an equivalence for Geronimus perturbations—the inverse operation of Christoffel transformations—in the setting of mixed multiple orthogonal polynomials. For a regular matrix polynomial $R(x)$ that is neither required to be monic nor restricted in the rank of its leading coefficient, and whose eigenvalues avoid the support of the measures, the perturbed orthogonality exists exactly when a family of tau-determinants $\\tau_n$ never vanishes. When that condition holds, the paper gives explicit Christoffel-type formulas: the perturbed type I and type II polynomials are written as determinants built from the original polynomials, their Cauchy transforms evaluated at the eigenvalues of $R$, and the free masses that define the perturbation. The same structure is developed for left multiplication, for eigenvalues of arbitrary multiplicity via Jordan chains, and for the Markov–Stieltjes matrix function, which transforms as $\\check{F}(z)=(F(z)+S(z))R^{-1}(z)$ with $\\deg S=\\deg R-1$. A worked three-weight Jacobi–Piñeiro example exhibits the formulas concretely.","feed_headline":"Perturbed orthogonality survives iff tau-determinants stay nonzero","feed_subtitle":"Explicit Christoffel-type formulas link old and new polynomials once every tau_n is nonzero.","key_machinery":"The load-bearing object is the tau-determinant $\\tau_n$, the determinant of the $M\\times M$ linear system (with $M=Np-r$) that determines the unknown entries of the connection matrix $\\Omega=\\check{S}S^{-1}$. The columns of this system are the vectors $D^{(i)}_m-W^{(i)}_m$, where $D^{(i)}_m$ are Cauchy transforms of the original type II polynomials evaluated at the eigenvalue $x_i$ of $R(x)$ and $W^{(i)}_m$ are the corresponding contributions of the delta masses; for multiple eigenvalues these become row vectors of length equal to the partial multiplicities, using Jordan chains. Nonzero $\\tau_n$ makes the system invertible, yielding $\\Omega_{n,n-M}=(-1)^M\\tau_n/\\tau_{n-1}$ and the other connection coefficients as ratios of tau-determinants, which then feed into determinantal Christoffel-type formulas for $\\check{A}$ and $\\check{B}$. The companion machinery is the matrix Christoffel–Darboux kernel $K_D^{[n]}$, the mixed kernel $K^{[n],(i)}$, and the banded recurrence matrix $T$, together with the divisibility theory of matrix polynomials for the Jordan-chain generalization.","core_discovery":"The central claim, stated as Theorems 2.12, 2.13, 3.1 and 3.5, is that for a Geronimus perturbation $d\\check{\\mu}(x)=d\\mu(x)R^{-1}(x)$ plus delta masses at the eigenvalues of $R$, perturbed mixed multiple orthogonality exists if and only if $\\tau_n\\ne 0$ for every $n\\in\\mathbb{N}_0$. The tau-determinants are built from the data $D^{(i)}_n-W^{(i)}_n$: Cauchy transforms of the original polynomials evaluated at the eigenvalues of $R$, corrected by contributions of the added masses. When the determinants are nonzero, the connection matrix $\\Omega$ between the perturbed and original Gauss–Borel factorizations is lower unitriangular with entries given by ratios of tau-determinants, and the perturbed type II polynomials are explicit $(Np-r+1)\\times(Np-r+1)$ determinants whose last column contains $B^{(b)}_{n-M}(x),\\dots,B^{(b)}_n(x)$, while the perturbed type I polynomials are expressed through the mixed Christoffel–Darboux kernels and the inverse of the linear system. The proof of necessity uses a contour-integral argument: a vanishing $\\tau_n$ forces a nonzero vector that is annihilated by the eigenvectors, a contradiction. The proof of sufficiency shows that nonzero $\\tau_n$, together with the first $Np-r$ orthogonality conditions, propagate the biorthogonality and degree structure for all $n$. For eigenvalues of higher multiplicity, eigenvectors are replaced by canonical sets of Jordan chains, and the left-multiplication version swaps the roles of type I and type II and of left and right eigenvectors.","pith_inferences":["The tau-determinant criterion should compose: applying a Geronimus perturbation followed by a Christoffel perturbation (a Uvarov transformation) should produce a combined criterion combining the tau-determinants of this paper with those of the companion Christoffel paper, since both transformations act on the same Gaussian factorization.","The equivalence gives a practical detection tool for spectral algorithms: in numerical constructions of banded recurrence matrices, the index at which a tau-determinant crosses zero marks exactly where the attempted perturbed orthogonal family stops being orthogonal.","For scalar weights ($p=q=1$), the paper's machinery should reproduce the classical Geronimus formulas with free parameters, which would provide a simple consistency check for the determinant formulas.","The matrix linear spectral transformation form suggests an interpretation of these perturbations as finite-rank updates of the resolvent: $\\check{F}(z)R(z)=F(z)+S(z)$ resembles a finite-rank perturbation of the Stieltjes transform, which may connect to operator-theoretic treatments."],"forward_implications":["Geronimus-perturbed orthogonality is controlled by an explicit sequence: if any $\\tau_n$ vanishes, no orthogonality exists beyond the first $Np-r$ degrees; if none vanish, the whole biorthogonal family exists.","The Christoffel-type formulas give a direct algorithm: compute the Cauchy data $D^{(i)}_m$ and mass corrections $W^{(i)}_m$, form the tau-determinants, and obtain the perturbed polynomials by determinant expansions—no need to solve for the full connection matrix.","Right and left Geronimus perturbations are dual: formulas for $\\check{B}$ under right multiplication become formulas for $\\check{A}$ under left multiplication, with left and right eigenvectors exchanged.","The Markov–Stieltjes matrix function of the perturbed measure is a matrix linear spectral transformation $\\check{F}(z)=(F(z)+S(z))R^{-1}(z)$ with $\\deg S=\\deg R-1$, extending Zhedanov's scalar rational spectral transformation picture to the mixed multiple setting.","In the Jacobi–Piñeiro three-weight case, the general formulas reduce to explicit determinants involving endpoint values of the classical polynomials and two free mass parameters $\\xi_0,\\xi_1$."],"supporting_citations":[{"why":"The authors' companion Christoffel perturbation paper whose Gauss–Borel and tau-determinant framework is adapted here to the Geronimus (inverse) transformation.","marker":"[40]"},{"why":"Supplies the Smith form, Jordan chain, and divisibility theory for matrix polynomials used to treat eigenvalues of arbitrary multiplicity.","marker":"[34]"},{"why":"Provides the explicit Jacobi–Piñeiro polynomials, recurrence coefficients, and endpoint evaluations used in the three-weight case study.","marker":"[16]"},{"why":"Establishes the Gauss–Borel factorization viewpoint for mixed-type multiple orthogonal polynomials that underlies the moment-matrix setup.","marker":"[6]"},{"why":"Earlier work on matrix Geronimus transformations with spectral techniques for monic perturbations; this paper generalizes its connection-matrix approach to non-monic, arbitrary-rank cases.","marker":"[7]"},{"why":"Gives prior Christoffel and Geronimus formulas for two-weight non-mixed multiple orthogonal polynomials, the setting this paper extends.","marker":"[17]"},{"why":"Defines the scalar linear spectral transformations of Stieltjes functions that the Markov–Stieltjes result is compared with.","marker":"[49]"}],"fun_headline_variants":["Geronimus perturbations: orthogonality hinges on nonzero tau","Tau determinants decide Geronimus perturbed orthogonality","Geronimus: orthogonality iff tau-determinants survive","Mixed multiple orthogonality: Geronimus requires nonzero tau","Geronimus perturbed orthogonality: tau must stay nonzero"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The matrix polynomial $R(x)$ must be invertible at every point of the support of the original measures, so that the regular part $d\\mu R^{-1}$ in the perturbed measure has no poles on the support.","fun_headline_variants_meta":{"raw":{"variants":["Geronimus perturbations: orthogonality hinges on nonzero tau","Tau determinants decide Geronimus perturbed orthogonality","Geronimus: orthogonality iff tau-determinants survive","Mixed multiple orthogonality: Geronimus requires nonzero tau","Geronimus perturbed orthogonality: tau must stay nonzero"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000866,"raw_usage":{"total_tokens":3794,"prompt_tokens":1025,"completion_tokens":2769,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":641,"completion_tokens_details":{"reasoning_tokens":2681}},"tokens_in":641,"tokens_out":2769,"duration_ms":16591,"temperature":1.0,"reasoning_tokens":2681,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:38:06.928202+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The cleanest test is the scalar case $p=q=1$ with $R(x)=x-a$ for $a$ outside the support: the Geronimus measure is $d\\check{\\mu}=d\\mu/(x-a)+\\xi\\,\\delta(x-a)$. Compute the leading principal minors of the perturbed moment matrix and compare their vanishing with the tau-determinants $\\tau_n$ defined in Definition 2.8 (here $M=1$, so each $\\tau_n$ is the single entry $D_n(a)-W_n(a)$ up to normalization). Any mismatch between a zero $\\tau_n$ and a singular perturbed moment submatrix would disprove the claimed equivalence; the paper predicts exact coincidence.","supporting_citations":[{"cited_title":"General Christoffel Perturbations for Mixed Multiple Orthogonal Polynomials","cited_arxiv_id":"2405.11630","evidence_quote":"The authors' companion Christoffel perturbation paper whose Gauss–Borel and tau-determinant framework is adapted here to the Geronimus (inverse) transformation."},{"cited_title":"Gohberg, P","cited_arxiv_id":null,"evidence_quote":"Supplies the Smith form, Jordan chain, and divisibility theory for matrix polynomials used to treat eigenvalues of arbitrary multiplicity."},{"cited_title":"Classical multiple orthogonal polynomials for arbitrary number of weights and their explicit representation","cited_arxiv_id":"2404.13958","evidence_quote":"Provides the explicit Jacobi–Piñeiro polynomials, recurrence coefficients, and endpoint evaluations used in the three-weight case study."},{"cited_title":"Álvarez-Fernández, U","cited_arxiv_id":null,"evidence_quote":"Establishes the Gauss–Borel factorization viewpoint for mixed-type multiple orthogonal polynomials that underlies the moment-matrix setup."},{"cited_title":"Ariznabarreta, J","cited_arxiv_id":null,"evidence_quote":"Earlier work on matrix Geronimus transformations with spectral techniques for monic perturbations; this paper generalizes its connection-matrix approach to non-monic, arbitrary-rank cases."},{"cited_title":"Branquinho, A","cited_arxiv_id":null,"evidence_quote":"Gives prior Christoffel and Geronimus formulas for two-weight non-mixed multiple orthogonal polynomials, the setting this paper extends."},{"cited_title":"Zhedanov, Rational spectral transformations and orthogonal polynomials, Journal of Computational Applied Math- ematics 85 (1997) 67-86","cited_arxiv_id":null,"evidence_quote":"Defines the scalar linear spectral transformations of Stieltjes functions that the Markov–Stieltjes result is compared with."}],"review_version":1}