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Multiple analogues of binomial coefficients and related families of special numbers [article]

Hasan Coskun
<span title="2010-01-20">2010</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We construct multiple qt-binomial coefficients and related multiple analogues of several celebrated families of special numbers in this paper.  ...  These multidimensional generalizations include the first and the second kind of qt-Stirling numbers, qt-Bell numbers, qt-Bernoulli numbers, qt-Catalan numbers and the qt--Fibonacci numbers.  ...  Multiple basic and ordinary special numbers In this section we give multiple basic and multiple ordinary analogues (or the qt-and α-analogues, respectively) of several celebrated families of special numbers  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1001.3466v1">arXiv:1001.3466v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/psb7cxfyr5gl5ixxzzyd3rnbki">fatcat:psb7cxfyr5gl5ixxzzyd3rnbki</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-1001.3466/1001.3466.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/03/d5/03d51ec7c96f699204468ab20b644a6dce1f732c.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1001.3466v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Multiple analogues of binomial coefficients and families of related special numbers

Hasan Coskun
<span title="">2010</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Multiple qt-binomial coefficients and multiple analogues of several celebrated families of related special numbers are constructed in this paper.  ...  These higher-dimensional generalizations include the first and the second kind of qt-Stirling numbers, qt-Bell numbers, qt-Bernoulli numbers, qt-Catalan numbers and the qt-Fibonacci numbers.  ...  Multiple basic and ordinary special numbers In this section we give multiple basic and multiple ordinary analogues (or the qt-and α-analogues, respectively) of several celebrated families of special numbers  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2010.05.001">doi:10.1016/j.disc.2010.05.001</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/x67ueunxoncurnrqpi6sbfkgzy">fatcat:x67ueunxoncurnrqpi6sbfkgzy</a> </span>
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Page 234 of Mathematical Reviews Vol. , Issue 2003A [page]

<span title="">2003</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The numbers in the (L + 1)th row are the coefficients of (1+.x +x)" which gives representations as binomial sums. In their joint work on extensions of the hard-hexagon model G. E. Andrews and R. J.  ...  A related and simpler model of the first-order linear equation yields the various q-analogues of the exponential function.” See also «01024, 05148, 05149, 11044, 16044, 43008, 81073, 81075, 82020.  ... 
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Multiple bracket function, Stirling number, and Lah number identities

Hasan Coskun
<span title="">2018</span> <i title="International Press of Boston"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ppxcitfjyjcpjl6wlzzfryrguu" style="color: black;">Journal of Combinatorics</a> </i> &nbsp;
The author has constructed multiple analogues of several families of combinatorial numbers in a recent article, including the bracket symbol, and the Stirling numbers of the first and second kind.  ...  In the present paper, a multiple analogue of another sequence, the Lah numbers, is developed, and certain associated identities and significant properties of all these sequences are constructed.  ...  The multiple qt-binomial coefficients The multiple Stirling numbers we develop in this paper are closely connected with binomial coefficients just as in the one dimensional case.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4310/joc.2018.v9.n3.a5">doi:10.4310/joc.2018.v9.n3.a5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4pbay6fem5b7nkpc4l6asruh3i">fatcat:4pbay6fem5b7nkpc4l6asruh3i</a> </span>
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Page 3316 of Mathematical Reviews Vol. , Issue 96f [page]

<span title="">1996</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The author considers a finite field analogue of the binomial co- efficient, defined, for two multiplicative characters, in terms of a Jacobi sum.  ...  He uses a distribution relation of Davenport-Hasse for Gauss sums, and a classical relation between Gauss and Jacobi sums, to deduce an identity between certain products of these bi- nomial coefficient  ... 
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Multiple Bracket Function, Stirling Number, and Lah Number Identities [article]

Hasan Coskun
<span title="2015-06-17">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The author has constructed multiple analogues of several families of combinatorial numbers in a recent article, including the bracket symbol, and the Stirling numbers of the first and second kind.  ...  In the present paper, a multiple analogue of another sequence, the Lah numbers, is developed, and certain associated identities and significant properties of all these sequences are constructed.  ...  functions they satisfy, and their connections to different families of multiple combinatorial numbers in an upcoming article.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1212.6573v2">arXiv:1212.6573v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wbpa7ez3zffldalpamfleg6rqe">fatcat:wbpa7ez3zffldalpamfleg6rqe</a> </span>
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Page 1718 of Mathematical Reviews Vol. , Issue 87d [page]

<span title="">1987</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Hence, by specializing the values of p;, p2, qi, 92 in the recurrence relation for G(n,m), a number of summation identities involving these special sequences or polynomials are obtained.  ...  The author addresses the problem of computing any number n, starting with the number 1, through a sequence of additions and multiplications of previously obtained numbers.  ... 
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Combinatorial Formulas for Certain Sequences of Multiple Numbers [article]

Hasan Coskun
<span title="2016-01-01">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Multiple analogues of certain families of combinatorial numbers are recently constructed by the author in terms of well poised Macdonald functions, and some of their fundamental properties are developed  ...  In this paper, we present combinatorial formulas for the well poised Macdonald functions, the multiple binomial coefficients, the multiple bracket function, and the multiple Catalan and Lah numbers.  ...  The classical Catalan numbers are defined by the relation C n := 1 n + 1 2n n (51) A multiple qt-analogue of these numbers is defined in terms of the multiple binomial coefficients and the factorial functions  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1601.00052v1">arXiv:1601.00052v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tnqyr7o2ozh4xhexbumtps2zau">fatcat:tnqyr7o2ozh4xhexbumtps2zau</a> </span>
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Elliptic Analogues of the Macdonald and Koornwinder Polynomials

Eric M. Rains
<span title="">2011</span> <i title="Published by Hindustan Book Agency (HBA), India. WSPC Distribute for All Markets Except in India"> Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) </i> &nbsp;
In recent work, the author has constructed elliptic analogues: a family of multivariate functions on an elliptic curve satisfying analogues of the Macdonald conjectures, and degenerating to Macdonald and  ...  Perhaps the nicest multivariate orthogonal polynomials are the Macdonald and Koornwinder polynomials, respectively 2-parameter deformations of Schur functions and 6-parameter deformations of orthogonal  ...  A final collection of open problems is a family of conjectured quadratic transformations which were stated (together with proofs of a number of special cases and internal consistency checks) in [20] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1142/9789814324359_0157">doi:10.1142/9789814324359_0157</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vd6m2tdplrbnrnm6bnbrmejqfi">fatcat:vd6m2tdplrbnrnm6bnbrmejqfi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170809004859/http://www.mathunion.org/ICM/ICM2010.4/Main/icm2010.4.2530.2554.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/65/8b/658b850a0143de822940a6a57475d0ca49841b36.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1142/9789814324359_0157"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Page 2518 of Mathematical Reviews Vol. , Issue 2003d [page]

<span title="">2003</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
In 1917 Schur introduced a poly- nomial analogue of these identities, where the sum is replaced by a sum over g-binomial coefficients and the product becomes an alternating sum of g-binomial coefficients  ...  (English summary) Symbolic computation, number theory, special functions, physics and combinatorics (Gainesville, FL, 1999), 255-265, Dev. Math., 4, Kluwer Acad. Publ., Dordrecht, 2001.  ... 
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Page 4460 of Mathematical Reviews Vol. , Issue 95h [page]

<span title="">1995</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
95h:05009 s = 2; using this, various enumeration formulas involving Catalan numbers and binomial coefficients are derived.  ...  Special cases involve the Fibonacci numbers and some of their generalizations. By counting paths in different ways, the authors prove identities for sums of binomial and multinomial co- efficients.  ... 
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Catalan triangle numbers and binomial coefficients [article]

Kyu-Hwan Lee, Se-jin Oh
<span title="2017-10-17">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac--Moody algebras.  ...  We prove that any binomial coefficient can be written as weighted sums along rows of the Catalan triangle.  ...  Catalan expansion of binomial coefficients In this section, we prove expressions of binomial coefficients as 2-power weighted sums along rows of the Catalan triangle.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1601.06685v2">arXiv:1601.06685v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/yzkk3se7ojb2bhntq6hqbnjz3e">fatcat:yzkk3se7ojb2bhntq6hqbnjz3e</a> </span>
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THE TOTAL SIZE OF A GENERAL STOCHASTIC EPIDEMIC

NORMAN T. J. BAILEY
<span title="">1953</span> <i title="Oxford University Press (OUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/6oeltljhrzfq7brjdtp2wqehpu" style="color: black;">Biometrika</a> </i> &nbsp;
N is the total number of families of a given size; and a, is the observed number of families with a total of w cases in addition to the first one. The case n = 0 is trivial.  ...  The probability approach was fundamental to Greenwood's (1931 Greenwood's ( , 1946) ) use of chain binomials in discussing the distribution of multiple cases of disease in households.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1093/biomet/40.1-2.177">doi:10.1093/biomet/40.1-2.177</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hawsvu727fhd7ioeagkt2ndlnm">fatcat:hawsvu727fhd7ioeagkt2ndlnm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170810065153/http://www.epi.msu.edu/janthony/requests/articles/bailey_total%20size%20general%20stochastic%20epidemic.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/02/e4/02e46709ce6077668ff477e3a527b093bc5dffdd.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1093/biomet/40.1-2.177"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> oup.com </button> </a>

The Total Size of a General Stochastic Epidemic

Norman T. J. Bailey
<span title="">1953</span> <i title="JSTOR"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/6oeltljhrzfq7brjdtp2wqehpu" style="color: black;">Biometrika</a> </i> &nbsp;
N is the total number of families of a given size; and a, is the observed number of families with a total of w cases in addition to the first one. The case n = 0 is trivial.  ...  The probability approach was fundamental to Greenwood's (1931 Greenwood's ( , 1946 use of chain binomials in discussing the distribution of multiple cases of disease in households.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2307/2333107">doi:10.2307/2333107</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/y3m3d4ndffdqla3errjjn3kxte">fatcat:y3m3d4ndffdqla3errjjn3kxte</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170810065153/http://www.epi.msu.edu/janthony/requests/articles/bailey_total%20size%20general%20stochastic%20epidemic.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/02/e4/02e46709ce6077668ff477e3a527b093bc5dffdd.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2307/2333107"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> jstor.org </button> </a>

The negative q-binomial [article]

Shishuo Fu, Victor Reiner, Dennis Stanton, Nathaniel Thiem
<span title="2011-08-23">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Interpretations for the q-binomial coefficient evaluated at -q are discussed. A (q,t)-version is established, including an instance of a cyclic sieving phenomenon involving unitary spaces.  ...  Acknowledgements The authors thank Paul Garrett, John Shareshian, Eric Sommers, and Ryan Vinroot for helpful comments.  ...  Our main result Theorem 1 is an analogue of (1.2) for the primed q-binomial coefficient that clearly demonstrates (2.2) .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1108.4702v1">arXiv:1108.4702v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/maspl5daxzbr5kohnhbhsakeuq">fatcat:maspl5daxzbr5kohnhbhsakeuq</a> </span>
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