Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Zeros of linear combinations of Laguerre polynomials

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A single auxiliary polynomial Q or P determines, for large n, how many real zeros a finite consecutive Laguerre combination has.

desk verdict A solid, substantial contribution to the zero theory of finite Laguerre sums; the main theorems look right, but the monic-case bridge identity is not fully proved as written and one remark contains a sign error. read the letter →

arxiv 2507.22425 v1 pith:ALGGIJNY submitted 2025-07-30 math.CA

Antonio J. Durán This is my paper · ORCID
classification math.CA MSC 42C0526C1033C45
keywords zerosLaguerrepolynomialslinearcombinationsreal-rootednessquasi-spectralpropertiesgeneralizedBellinterlacingBrenke
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that for a finite combination $q_n(x)=\sum_{j=0}^K \gamma_j \tilde L_{n-j}^\alpha(x)$ of $K+1$ consecutive Laguerre polynomials, the eventual number and location of real zeros is read off from a single auxiliary polynomial: $Q(x)=\sum_{j=0}^K(-1)^j\gamma_j(x)_{K-j}$ for the monic normalization and $P(x)=\sum_{j=0}^K\gamma_j x^{K-j}$ for the other three normalizations. If $Q$ has only real zeros, all zeros of $q_n$ are real and simple for $n\ge n_1$ and positive for $n\ge n_0$, with explicit thresholds computed from the zeros of $Q$; if $Q$ has non-real zeros, the monic $q_n$ still has only positive simple zeros for large $n$. In the other three cases, if $P$ has only real zeros then $q_n$ has only real zeros, and if $P$ has $m$ non-real zeros then exactly $m$ non-real zeros persist for large $n$. The paper also shows which of the four normalizations preserve real-rootedness under the map $T(x^n)=p_n$, and it recovers and sharpens earlier $n=K$ criteria. These criteria matter because earlier results guaranteed only that such combinations eventually have many real zeros; here the whole zero configuration is settled by a finite polynomial and explicit degree bounds.

What carries the argument

The engine is a set of quasi-spectral identities: first- or second-order differential operators act on each normalized Laguerre family as one-step shifts with constant eigenvalue, for example $\Lambda_\alpha(\hat L_n^\alpha)=-\hat L_{n+1}^{\alpha-1}$ for the monic family, $\Upsilon_\alpha(\mathcal L_n^\alpha)=\mathcal L_n^{\alpha-1}$ for the value-one-at-zero family, and $\Omega_\alpha$ as the corresponding shift for the Brenke family. For the monic case, iterating the backward shift connects $q_n$ to generalized Bell polynomials $b_n^{r;\phi}$ defined by the recurrence $b_{n+1}=\Lambda_r b_n+\phi_{n+1}b_n$, where $\phi$ is built from the zeros of $Q$; the bridge identity $q_n(x)=(-1)^n b_n^{\alpha+n-K+1;\phi}(x)$ (Lemma 3.4) transfers zero-counting to a theorem about zeros of these Bell-type polynomials. For the other normalizations, the differential operators are complex zero decreasing, and Laguerre asymptotics expressed through the conformal map $\varphi(z)=\tfrac12(z-2+\sqrt{z^2-4z})$ convert the limiting zero set of $q_n$ into the zero set of $P(-\varphi(z))$.

What would settle it

Take the sharpest monic threshold: $\alpha=0$, $K=2$, $\theta_1=1/4$, $\theta_2=1/2$, which via (3.18)–(3.19) gives $\gamma_1=7/4$, $\gamma_2=1/8$; Theorem 1.3(1) predicts that every $q_n$, starting at $n_0=2$, has only positive simple zeros, so computing the zeros of $q_2=\hat L_2^0+\frac74\hat L_1^0+\frac18\hat L_0^0$, and then of $q_3$ and $q_4$, either confirms the asserted thresholds or exhibits a concrete failure of the bridge identity's omitted induction.

Watch

Extended reading notes

Core claim

The central discovery is that the four normalizations separate into two regimes. For monic Laguerre polynomials, Theorem 1.3 proves that the zeros of $Q$ are the only data needed: if all zeros of $Q$ are real, every $q_n$ has real and simple zeros for $n\ge n_1$ and positive and simple zeros for $n\ge n_0$, with $n_l=\max\{K,\lfloor\theta_i-\alpha+K\rfloor:1\le i\le m-l\}$, and if $Q$ has non-real zeros, positivity still holds for large $n$. For the normalized families, Theorems 1.4 through 1.6 prove that $P$ controls the count: under the disk condition $P(z)\ne 0$ for non-real $|z|\le 1$, if $P$ has $N^{\rm nr}$ non-real zeros, then for large $n$ the polynomial $q_n$ has exactly $n-N^{\rm nr}$ real simple zeros, with $N^1$ of them negative, and the real zeros of consecutive $q_n$ interlace. In the Brenke normalization the statement becomes an equivalence: $q_n$ has only real zeros for all $n$ if and only if all zeros of $P$ are real; otherwise exactly $N^{\rm nr}$ non-real zeros persist. Thus normalization is not a cosmetic detail—it changes the answer, and the paper quantifies precisely how.

Load-bearing premise

The monic theorem depends on the bridge identity $q_n(x)=(-1)^n b_n^{\alpha+n-K+1;\phi}(x)$ holding for every $n\ge K$, but Lemma 3.4 verifies only $n=K$ and $n=K+1$ and asserts the rest "can be completed proceeding similarly"; if that omitted induction has a hidden case-dependence, Theorem 1.3 collapses, and the non-monic theorems additionally require the disk condition $P(z)\ne 0$ for non-real $|z|\le 1$.

Editorial extensions

If this is right

  • For monic combinations with $Q$ having only real zeros, all zeros of $q_n$ are real and simple once $n$ reaches an explicit bound computed from the large zeros of $Q$; when all zeros of $Q$ lie below $\alpha+1$, real-and-positive rootedness holds immediately for every $n\ge K$.
  • For the value-one-at-zero and standard Laguerre normalizations, if $P$ has no non-real zeros in the closed unit disk and $P(1)\ne 0$, the number of real zeros of $q_n$ is exactly $n-N^{\rm nr}$ for large $n$, with exactly $N^1$ negative zeros, and the real zeros of consecutive $q_n$ interlace.
  • For the Brenke normalization, real-rootedness of $q_n$ for all $n$ is equivalent to $P$ having only real zeros; otherwise exactly $N^{\rm nr}$ non-real zeros survive.
  • The $n=K$ case recovers and sharpens earlier criteria: when the auxiliary polynomial has only real zeros and the Laguerre parameter is in the stated range, expansions of the form $\sum_j \tau_j L_j^\alpha$ are real-rooted.
  • Corollary 1.7 gives new cases in which the linear operator $T(x^n)=p_n$ preserves real-rootedness, including all real $\alpha>-1$ for two of the normalizations and integer-$\alpha$ cases for the standard normalization.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same quasi-spectral mechanism should yield finite auxiliary polynomials for other classical or semi-classical families with an explicit backward-shift operator, giving exact thresholds rather than asymptotic statements.
  • The explicit $n_0$ and $n_1$ formulas turn the monic theorem into a finite algebraic certificate: test the auxiliary polynomial for real roots, then a finite computation validates all degrees.
  • The Brenke equivalence suggests that any failure of real-rootedness in that normalization must come from the non-real zeros of $P$, not from the Bessel-type factor in the generating function, which could guide searches for real-rooted generating functions.
  • The contrast between the monic and Brenke normalizations indicates that normalization can be chosen deliberately when a finite combination is engineered to have prescribed real-zero behaviour, for example in quadrature or spectral constructions.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the real and non-real zeros of finite linear combinations of K+1 consecutive Laguerre polynomials in four normalizations: monic, value-one-at-zero, standard, and Brenke. The main results, Theorems 1.3–1.6, assert that for large n the zero behaviour of q_n is governed by the auxiliary polynomial Q(x)=Σ(-1)^j γ_j (x)_{K-j} in the monic case and by P(x)=Σ γ_j x^{K-j} in the other three cases. In particular, the paper gives conditions on the zeros of Q or P under which all zeros of q_n are real, simple, positive, or interlace, and, when P has non-real zeros, it describes exactly how many non-real zeros persist for large n. The proofs combine the quasi-spectral properties of Laguerre polynomials with generalized Bell-type polynomial families, complex-zero-decreasing operators, and asymptotic expansions.

Significance. If the main theorems are correct, this is a substantial contribution to the zero-location theory of Laguerre combinations and to the real-rootedness preservation literature. The paper improves earlier results of Iserles, Nørsett, and Saff, provides explicit thresholds n_0 and n_1, gives a counterexample showing that interlacing can fail for small n, and connects the problem to generalized Bell polynomials and Brenke polynomials. The four-normalization comparison is a useful unifying framework. The paper is generally detailed and mostly self-contained, and the statements of Theorems 1.3–1.6 are sharp and falsifiable.

major comments (3)
  1. [Section 3, Lemma 3.4] The bridge identity (3.11) is proved only for the case NA=0 and then for n=K and n=K+1; the text states that 'the proof for the rest of the cases can be completed proceeding similarly.' Since Theorem 1.3 uses (3.11) for every n ≥ K in Step 1 and uses (3.23) in Step 2, the monic-case theorem rests on an omitted induction. Please supply the full induction step for n ≥ K+2, or provide a precise reduction to the proved cases. The operator-index slip in this passage ('apply Λ_{α+2}' followed by Λ_{α+1} in the display) should also be corrected.
  2. [Section 5, Remark 5.2] Remark 5.2 is self-contradictory. It aims to show that Corollary 5.4 cannot be true for noninteger α < K-2, but after using [29, Theorem 6.73] to bound the number of real zeros of q_n by n + floor(α-K+1) + 1, it concludes 'and so Corollary 5.4 can be true.' The displayed inequality is also wrong: for noninteger α < K-2 one has floor(α-K+1)+1 ≤ -1, so q_n has fewer than n real zeros. The correct conclusion is that Corollary 5.4 cannot hold. This error affects Corollary 1.7(1) and the surrounding discussion.
  3. [Section 2, Lemma 2.6] Lemma 2.6 is stated with the proof omitted ('the proof is similar to the usual proof for Hurwitz's Theorem'). The lemma is used as a load-bearing tool in Step 6 of the proof of Theorem 1.4 and again in the interlacing argument in Theorem 1.5. Since it guarantees uniform persistence of non-real zeros, the proof or a precise reference should be included rather than left to the reader.
minor comments (5)
  1. [Section 5, Lemma 5.2] The proof of Lemma 5.2 is omitted with the explanation that it is the same as that of Lemma 4.3. This is acceptable only if the identical calculation is explicitly acknowledged; please include the statement or a precise pointer so the reader does not need to reconstruct the asymptotic.
  2. [Section 8, Corollary 8.2] The proof of Corollary 8.2 is omitted because it is similar to that of Corollary 8.1. Since this corollary is not central to the main theorems, a clear reference to Corollary 8.1 is enough, but the sentence should be completed with the relevant details or a citation.
  3. [Abstract and Section 1] The abstract states 'if P has m>1 non-real zeros', but the theorems treat the case N_nr > 0, i.e., m ≥ 1. Please correct the inequality.
  4. [Abstract] The phrase 'second, third and forth cases' contains a typo; it should be 'fourth'.
  5. [Section 3, Theorem 1.3] The notation in Theorem 1.3 uses m for the number of real zeros satisfying α+1 ≤ θ_j, while the proof of Step 1 uses r_1 for the same quantity. Please make the notation consistent to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the zero theorems are derived from explicit identities, asymptotics, and independent prior theorems; the flagged weaknesses are proof gaps, not circular reductions.

full rationale

No equation in the paper defines a predicted quantity as the same fitted input, and no theorem is invoked whose assumptions already contain the conclusion. The monic-case bridge identity (3.11) is a proved identity (modulo an omitted induction), not an ansatz or a definition of the target zeros. Step 2 of Theorem 1.3 uses Corollary 1.2 from the author's earlier paper [9]; this is a separately stated theorem with explicit hypotheses and an explicit n-bound, so under the stated rubric it counts as independent support rather than circular self-citation. Likewise, Theorem 1.6 imports [8, Cor. 6.1] and the other sections use external results (Craven--Csordas, Beardon--Driver, Hurwitz asymptotics), all with stated assumptions not containing the respective conclusions. The derivation chain is therefore not circular. Two non-circular correctness issues are flagged for completeness. First, Lemma 3.4 proves the bridge identity (3.11) only for n=K and n=K+1, then says 'The proof for the rest of the cases can be completed proceeding similarly' after (3.17); since Theorem 1.3 applies (3.11) for every n at least K, the monic theorem currently rests on an omitted induction. Second, Remark 5.2 ends with 'and so Corollary 5.4 can be true' immediately after proving the opposite for non-integer alpha < K-2, which appears to be a logical slip rather than a circular step. Neither issue makes an output equal to an input, so the circularity score remains 0.

Assumptions & free parameters 0 free parameters · 9 assumptions · 1 invented entities

The central theorems depend on standard Laguerre facts plus the author's earlier theorem used as Corollary 1.2; no parameters are fitted to data. The main unproved-in-text inputs are Lemma 2.6 and the incomplete induction in Lemma 3.4, which should be completed before the proof is fully self-contained.

assumptions (9)
  • domain assumption Corollary 1.2 of the author's prior paper [9]: for monic Laguerre combinations, q_n has only real and simple zeros for n at least an explicit bound.
    Used in Steps 2 and 4 of Theorem 1.3 as the black box that starts the bootstrap for non-real zeros of Q and for large shifts by upsilon.
  • standard math Quasi-spectral identities (1.7), (1.8), (1.9) and (5.3) for the four normalizations.
    Stated in Section 1 as known or straightforward; they are the reason all four normalization cases can be handled by differential operators.
  • standard math Laguerre asymptotic (4.8) of Geronimo-Van Assche, and Mehler-Heine formulas (8.4).
    Used to derive (4.11), (4.13), (5.6), (8.1), and to locate non-real zeros via Hurwitz's theorem.
  • standard math Beardon-Driver Theorem 2.4 on zeros of polynomials in the span of p_r through p_n.
    Gives at least n-K real zeros between Laguerre zeros and hence the bulk-zero placement used throughout Sections 4 to 6 and 8.
  • ad hoc to paper Lemma 2.6: analytic perturbation preserves N non-real zeros for large n, uniformly in lambda.
    Stated without proof in Section 2; used in Step 6 of Theorems 1.4 and 1.6 to obtain interlacing for large n.
  • ad hoc to paper Lemma 3.4 identity q_n(x)=(-1)^n b^{alpha+n-K+1;phi}_n(x), proved only up to n=K+1.
    The remaining induction is asserted, not written out; this identity is the bridge that lets Theorem 3.2 control q_n through Bell-type polynomials.
  • domain assumption Condition (1.15): P has no non-real zeros with |z| at most 1.
    Hypothesis of Theorems 1.4 and 1.5; the asymptotic root-counting argument breaks down if P has non-real zeros inside the closed unit disk.
  • domain assumption Parameter restriction alpha at least K-N_nr-1 in Theorem 1.5.
    Restricts the standard-Laguerre theorem; Remark 5.2 gives a family intended to show the bound is needed, though the written counterexample has a coefficient error.
  • standard math Multiplier-sequence characterization (Theorem 2.5) and Laguerre-Polya class facts, used in Section 7.
    Used in Corollary 1.7 and items L.2 to L.4 to translate real-rootedness preservation statements.
invented entities (1)
  • Generalized Bell-type polynomial family b_n^{r;phi,psi} and its specialization b_n^{r;phi}
    purpose: Auxiliary recurrence-defined polynomials that encode q_n in the monic case; their zero behavior under the backward-shift operator gives Theorem 1.3.
    The family is explicitly defined by recurrence (3.5), so it is not an unexplained postulate; however it has no empirical or falsifiable content outside the proof.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Zeros of linear combinations of Laguerre polynomials." pith.science (2026). https://pith.science/paper/ALGGIJNY

@misc{pith2026250722425,
  author       = {Pith},
  title        = {Pith review of: Zeros of linear combinations of Laguerre polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ALGGIJNY}},
  note         = {Machine review of arXiv:2507.22425}
}
abstract

We study the number of real zeros of finite combinations of $K+1$ consecutive normalized Laguerre polynomials of the form $$ q_n(x)=\sum_{j=0}^K\gamma_j\tilde L^\alpha_{n-j}(x),\quad n\ge K, $$ where $\gamma_j$, $j=0,\cdots ,K$, are real numbers with $\gamma_0=1$, $\gamma_K\not =0$. We consider four different normalizations of Laguerre polynomials: the monic Laguerre polynomials $\hat L_n^\alpha$, the polynomials $\mathcal L_n^\alpha=n!L_n^\alpha/(1+\alpha)_n$ (so that $\mathcal L_n^\alpha(0)=1$), the standard Laguerre polynomials $(L_n^\alpha)_n$ and the Brenke normalization $L_n^\alpha/(1+\alpha)_n$. We show the key role played by the polynomials $Q(x)=\sum_{j=0}^K(-1)^j\gamma_j(x)_{K-j}$ and $P(x)=\sum_{j=0}^K\gamma_jx^{K-j}$ to solve this problem: $Q$ in the first case and $P$ in the second, third and forth cases. In particular, in the first case, if all the zeros of the polynomial $Q$ are real and less than $\alpha+1$, then all the zeros of $q_n$, $n\ge K$, are positive. In the other cases, if all the zeros of $P$ are real then all the zeros of $q_n$, $n\ge K$, are also real. If $P$ has $m>1$ non-real zeros, there are important differences between the four cases. For instance in the first case, $q_n$ has still only real zeros for $n$ big enough, but in the fourth case $q_n$ has exactly $m$ non-real zeros for $n$ big enough.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Zeros of GKP sequences of polynomials

    math.GM 2026-07 accept novelty 6.0 of 10

    Under |φ_n|+ψ_n<0, GKP polynomials have real simple interlacing zeros between the roots of the driving quadratic, with explicit extreme-zero asymptotics when ψ is constant.

Reference graph

Works this paper leans on

29 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [1]

    Ahlfors, Complex analysis

    L.V. Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics, McGraw-Hill, 1978

    work page 1978

  2. [2]

    Beardon, K.A

    A.F. Beardon, K.A. Driver, The zeros of linear combinations of orthogonal polynomials, J. Approx. Theory 137 (2005) 179–186

    work page 2005

  3. [3]

    Cardon, E.L

    D.A. Cardon, E.L. Sorensen and J.C. White, Interlacing properties of coefficient polynomials in differential operator representations of real-root preserving linear transformations, Constr. Approx. 57 (2023), 235–253

    work page 2023

  4. [4]

    Chasse, Linear Preservers and Entire Functions with Restricted Zero Loci, Univer- sity of Hawai’i, PhD dissertation (2011)

    M. Chasse, Linear Preservers and Entire Functions with Restricted Zero Loci, Univer- sity of Hawai’i, PhD dissertation (2011)

    work page 2011

  5. [5]

    Craven and G

    T. Craven and G. Csordas, On a converse of Laguerre’s theorem, ETNA 5 (1997), 7–17

    work page 1997

  6. [6]

    Driver, K

    K. Driver, K. Jordaan and N. Mbuyi, Interlacing of zeros of linear combinations of classical orthogonal polynomials from different sequences, Appl. Numer. Math. 59 (2009), 2424–2429

    work page 2009

  7. [7]

    Dur´ an, Generalized Bell polynomials, J

    A.J. Dur´ an, Generalized Bell polynomials, J. Approx. Theory 306 (2025), 106121

    work page 2025

  8. [8]

    Brenke polynomials with real zeros and the Riemann Hypothesis

    A.J. Dur´ an, Brenke polynomials with real zeros and the Riemann Hypothesis, arXiv:2405.18940 [math.CA]

    work page Pith review arXiv

Show all 29 references
  1. [9]

    Dur´ an, Zeros of linear combinations of orthogonal polynomials arXiv:2505.11956 [math.CA]

    A.J. Dur´ an, Zeros of linear combinations of orthogonal polynomials arXiv:2505.11956 [math.CA]

    arXiv

  2. [10]

    Dur´ an, Zeros of linear combinations of Hermite polynomials arXiv:2505.15330 [math.CA]

    A.J. Dur´ an, Zeros of linear combinations of Hermite polynomials arXiv:2505.15330 [math.CA]

    arXiv

  3. [11]

    Erd´ elyi, W

    A. Erd´ elyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, (Bateman project), Higher Trascen- dental Functions, Volumed I and II McGraw Hill, New York, 1953

    1953

  4. [12]

    Fisk, The Laguerre Polynomials Preserve Real-Rootedness, https://arxiv.org/abs/0808.2635 (2008)

    S. Fisk, The Laguerre Polynomials Preserve Real-Rootedness, https://arxiv.org/abs/0808.2635 (2008)

    2008 arXiv

  5. [13]

    Gawronski, On the asymptotic distribution of the zeros of Hermite, Laguerre, and Jon- qui´ ere polynomials, J

    W. Gawronski, On the asymptotic distribution of the zeros of Hermite, Laguerre, and Jon- qui´ ere polynomials, J. Approx. Theory 50 (1987), 214–231

    1987

  6. [14]

    Geronimo, W

    J.S. Geronimo, W. Van Assche, Relative asymptotics for orthogonal polynomials with un- bounded recurrence coefficients, J. Approx. Theory 62 (1990), 47–69

    1990

  7. [15]

    Householder, The numerical treatment of a single nonlinear equation, McGraw-Hill, New York, 1970

    A.S. Householder, The numerical treatment of a single nonlinear equation, McGraw-Hill, New York, 1970

    1970

  8. [16]

    Iserles and S.P

    A. Iserles and S.P. Nørsett, Zeros of transformed polynomials, SIAM J. Math. Anal. 21 (1990), 483–509

    1990

  9. [17]

    Iserles, S.P

    A. Iserles, S.P. Nørsett and E. B. Saff, On transformations and zeros of polynomials, Rocky Mt. J. Math. 21 (1991), 331–357. 38 ANTONIO J. DUR ´AN

    1991

  10. [18]

    Iserles and E

    A. Iserles and E. B. Saff, Zeros of expansions in orthogonal polynomials, Math. Proc. Camb. Phil. Soc. 105 (1989), 559-573

    1989

  11. [19]

    H. Ki, Y.O. Kim, On the zeros of some generalized hypergeometric functions, J. Math. Anal. Appl. 243 (2000), 249–260

    2000

  12. [20]

    Koelink, P

    E. Koelink, P. Rom´ an, W. Zudilin, A partial-sum deformation for a family of orthogonal polynomials, Preprint arXiv:2409.00261 [math.CA] (2024), 18 pages

    2024

  13. [21]

    Mez˝ o, On the maximum ofr-Stirling numbers, Adv

    I. Mez˝ o, On the maximum ofr-Stirling numbers, Adv. Appl. Math. 41 (2008), 293–306

    2008

  14. [22]

    Mez˝ o and R.B

    I. Mez˝ o and R.B. Corcino, The estimation of the zeros of the Bell and r-Bell polynomials, Appl. Math. Comput. 250 (2015), 727–732

    2015

  15. [23]

    Mez˝ o and J.L

    I. Mez˝ o and J.L. Ram ´ ırez, Divisibility properties of ther-Bell numbers and polynomials, J. Number Theory 177 (2017), 136–152

    2017

  16. [24]

    Peherstorfer, Linear combination of orthogonal polynomials generating positive quadrature formulas, Math

    F. Peherstorfer, Linear combination of orthogonal polynomials generating positive quadrature formulas, Math. Comput. 55 (1990) 231–241

    1990

  17. [25]

    Peherstorfer, Zeros of linear combinations of orthogonal polynomials, Math

    F. Peherstorfer, Zeros of linear combinations of orthogonal polynomials, Math. Proc. Camb. Phil. Soc. 117 (1995) 533–544

    1995

  18. [26]

    Piotrowski, Linear Operators and the Distribution of Zeros of Entire Functions, University of Hawai’i, PhD Dissertation (2007)

    A. Piotrowski, Linear Operators and the Distribution of Zeros of Entire Functions, University of Hawai’i, PhD Dissertation (2007)

    2007

  19. [27]

    P´ olya and J

    G. P´ olya and J. Schur, ¨Uber zwei arten von faktorenfolgen in der theorie der algebraischen glichungen, Journal f¨ ur die reine und angewandte Mathematik 144 (1914), 89–113

    1914

  20. [28]

    Shohat, On mechanical quadratures, in particular, with positive coefficients, Trans

    J.A. Shohat, On mechanical quadratures, in particular, with positive coefficients, Trans. Amer. Math. Soc. 42 (1937) 461–496

    1937

  21. [29]

    Szeg¨ o, Orthogonal Polynomials

    G. Szeg¨ o, Orthogonal Polynomials. Fourth edition. American Mathematical Society, Collo- quium Publications, Vol. XXIII. American Mathematical Society, Providence, R.I., 1975. Departamento de An´alisis Matem´atico and IMUS, Universidad de Sevilla, 41080 Sevilla, Spain Email a...

    1975

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.