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A kind of orthogonal polynomials and related identities II

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Zhi-Hong Sun

Classical Analysis and ODEs · Submitted 16 Nov 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim New formulas and recurrence relations for $d_n^{(r)}(x)$ are obtained, and a new proof of the formula for $d_n^{(r)}(x)^2$ is also given.

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arXiv ID1711.05985 (version 2)
Pith IDLAMHXPNE, Pith's permanent record of this paper and its review
CitationZhi-Hong Sun. "A kind of orthogonal polynomials and related identities II." arXiv:1711.05985 (2017). https://pith.science/paper/LAMHXPNE
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,853 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.25 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.93 · compares with prior results 0.37 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

For $n=0,1,2,\ldots$ let $d_n^{(r)}(x)=\sum_{k=0}^n\binom{x+r+k}k\binom{x-r}{n-k}$. In this paper we illustrate the connection between $\{d_n^{(r)}(x)\}$ and Meixner polynomials. New formulas and recurrence relations for $d_n^{(r)}(x)$ are obtained, and a new proof of the formula for $d_n^{(r)}(x)^2$ is also given. In addition, for $r>-\frac 12$ and $n\ge 2$ we show that $d_n^{(r)}(x)>\frac{(2x+1)^n}{n!}>0$ for $x>-\frac 12$, and $(-1)^nd_n^{(r)}(x)>0$ for $x<-\frac 12$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 For $n=0,1,2,\ldots$ let $d_n^{(r)}(x)=\sum_{k=0}^n\binom{x+r+k}k\binom{x-r}{n-k}$. background 0.87 0.50 0.03 asserted 0.82
s1 In this paper we illustrate the connection between $\{d_n^{(r)}(x)\}$ and Meixner polynomials. contribution 0.98 0.89 0.23 asserted 0.99
s2 New formulas and recurrence relations for $d_n^{(r)}(x)$ are obtained, and a new proof of the formula for $d_n^{(r)}(x)^2$ is also given. contribution 0.67 0.97 0.39 asserted 1.00
s3 In addition, for $r>-\frac 12$ and $n\ge 2$ we show that $d_n^{(r)}(x)>\frac{(2x+1)^n}{n!}>0$ for $x>-\frac 12$, and $(-1)^nd_n^{(r)}(x)>0$ for $x<-\frac 12$. result 0.99 0.95 0.31 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

For $n=0,1,2,\ldots$ let $d_n^{(r)}(x)=\sum_{k=0}^n\binom{x+r+k}k\binom{x-r}{n-k}$. In this paper we illustrate the connection between $\{d_n^{(r)}(x)\}$ and Meixner polynomials. New formulas and recurrence relations for $d_n^{(r)}(x)$ are obtained, and a new proof of the formula for $d_n^{(r)}(x)^2$ is also given. In addition, for $r>-\frac 12$ and $n\ge 2$ we show that $d_n^{(r)}(x)>\frac{(2x+1)^n}{n!}>0$ for $x>-\frac 12$, and $(-1)^nd_n^{(r)}(x)>0$ for $x<-\frac 12$.

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