### Citations

900 |
Basic Hypergeometric Series,
- Gasper, Rahman
- 2004
(Show Context)
Citation Context ... functions. The standard references for distance regular graphs and association schemes are [5] and [9], while [25] covers topics (1) and (2). The standard notation for hypergeometric series found in =-=[22]-=- is used here. 2. Association schemes and distance regular graphs. In classical coding theory the Krawtchouk polynomials are important for several problems. This was generalized to polynomials in P-po... |

377 |
An algebraic approach to the association schemes of coding theory,
- Delsarte
- 1973
(Show Context)
Citation Context ... distance regular graphs. In classical coding theory the Krawtchouk polynomials are important for several problems. This was generalized to polynomials in P-polynomial association schemes by Delsarte =-=[13]-=-, and in this section we review this basic setup. Recall that a graph G = (V; E) consists of a set V of vertices, and a set E of unordered pairs of elements of V , called the edges of G. If the graph ... |

234 |
Combinatorial Species and Tree-like Structures,
- Bergeron, Labelle, et al.
- 1997
(Show Context)
Citation Context ....2).sThis proof led to a combinatorial study of other classical polynomials: for example Laguerre, Jacobi, Meixner, [6], [11], [34], [35], [40], and to generalized versions of the exponential formula =-=[7]-=-. Foata and Garsia [19] generalized this proof to multilinear generating functions for Hermite polynomials. The Mehler formula is an example of a bilinear generating function, which naturally gives an... |

179 | Combinatorial aspects of continued fractions, Discrete Mathematics 41
- Flajolet
- 1980
(Show Context)
Citation Context ... [31] using these lattice paths for the qHermite polynomials. The idea is that the combinatorics of the paths replaces the analysis of integration. Viennot's theory was heavily influenced by Flajolet =-=[15]-=-,[16]. The linearization coefficients a(m; n; k) are defined by p n (x)p m (x) = m+n X k=jm\Gammanj a(m; n; k)p k (x); which is equivalent to a(m; n; k) = L(pm p n p k ) L(p k p k ) : There has been m... |

113 | Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array,
- Terwilliger
- 2005
(Show Context)
Citation Context ...], which says that any set of orthogonal polynomials whose duals (in the sense above) are orthogonal polynomials, must be special or limiting cases of the Askey-Wilson 4 OE 3 polynomials. Terwilliger =-=[39]-=- has a linear algebraic version of this theorem which hopefully will lead to an algebraic classification of the Q-polynomial distance regular graphs. A simple application of the graph structure to the... |

112 |
Une theorie combinatoire des polynomes orthogonaux,
- Viennot
- 1983
(Show Context)
Citation Context ...2 ): This is the third and last term in the exponential of (4.2).sThis proof led to a combinatorial study of other classical polynomials: for example Laguerre, Jacobi, Meixner, [6], [11], [34], [35], =-=[40]-=-, and to generalized versions of the exponential formula [7]. Foata and Garsia [19] generalized this proof to multilinear generating functions for Hermite polynomials. The Mehler formula is an example... |

107 | Advanced determinant calculus, Séminaire Lotharingien Combin - Krattenthaler |

103 | The complete intersection theorem for systems of finite sets,
- Ahlswede, Khachatrian
- 1997
(Show Context)
Citation Context ...tain a fixed t-element subset. If n ! n 0 (k; t), the bound (3.1) is not correct. For values of n in this range the correct bound was conjectured by Frankl [21] and proven by Ahlswede and Khachatrian =-=[1]. It -=-is based upon the family of subsets F i = fA 2 X k : jA " [t + 2i]jst + ig which clearly has the t-intersecting condition. Theorem A [1]. If F is a t-intersecting family, then jF jsmaxfjF i j : 0... |

82 |
Distance-regular graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 18. Berlin etc
- Brouwer, Cohen, et al.
- 1989
(Show Context)
Citation Context ...rch topics which are not discussed include group and algebra representations, tableaux and symmetric functions. The standard references for distance regular graphs and association schemes are [5] and =-=[9]-=-, while [25] covers topics (1) and (2). The standard notation for hypergeometric series found in [22] is used here. 2. Association schemes and distance regular graphs. In classical coding theory the K... |

66 |
Algebraic Combinatorics, Chapman and
- Godsil
- 1993
(Show Context)
Citation Context ...which are not discussed include group and algebra representations, tableaux and symmetric functions. The standard references for distance regular graphs and association schemes are [5] and [9], while =-=[25]-=- covers topics (1) and (2). The standard notation for hypergeometric series found in [22] is used here. 2. Association schemes and distance regular graphs. In classical coding theory the Krawtchouk po... |

56 |
Association schemes and coding theory
- Delsarte, Levenshtein
- 1998
(Show Context)
Citation Context ...lities in (3.10), the objective function is ∑ i∈M ai, subject to a0 = 1, ai ≥ 0, i ∈M − {0}. For e-error correcting codes one considers the choice M = {0, 2e+ 1, · · · , d}, there is large literature =-=[14]-=- on this problem in H(N, q) using Krawtchouk polynomials. Many such questions remain open for the other known infinite families of distance regular graphs. There are theorems relating zeros of the pol... |

53 |
p, q-Stirling numbers and set partition statistics,
- Wachs, White
- 1991
(Show Context)
Citation Context ...the q-Charlier polynomialssn = n X k=1 S q (n; k)a k ; the Viennot theory offers its own statistic, which is not obviously q-Stirling distributed. Considering these two statistics led Wachs and White =-=[41]-=- to new bivariate equidistribution theorems for these two statistics. There are also many relations to continued fractions [15], [16] and determinants, via the moment generating function [28] and Hank... |

41 |
The combinatorics of q-Hermite polynomials and the Askey-Wilson integral
- Ismail, Stanton, et al.
- 1987
(Show Context)
Citation Context ...integral. One may ask if a q-version of Foata's proof of the Mehler's proves (4.4). One can interpret the q-Hermite polynomials as generating functions of involutions with a q-statistic (see [24] and =-=[31]-=-). A combinatorial version of the q-exponential formula [24] establishes the q-Hermite generating function, although it does not prove the q-Mehler formula (4.4). Hung Ngo [37] found a Foata-style pro... |

33 |
Restricted growth functions, rank row matchings of partition lattices, and q-Stirling numbers.
- Milne
- 1982
(Show Context)
Citation Context ...h positive integer coefficients which sum to S(n; k). Thus it is natural to consider S q (n; k) as a generating function for a statistic on set partitions of [n] into k blocks. This was done by Milne =-=[36]-=-. However, since S q (n; k) appear naturally in the moments of the q-Charlier polynomialssn = n X k=1 S q (n; k)a k ; the Viennot theory offers its own statistic, which is not obviously q-Stirling dis... |

31 |
The exact bound in the Erdős–Ko–Rado theorem, Combinatorica 4
- Wilson
- 1984
(Show Context)
Citation Context ...s widely known application to extremal set theory, the Erdos-Ko-Rado theorem, will be given. It uses the eigenvalues of the J(n; k) which are dual Hahn polynomials (2.3b). This proof is due to Wilson =-=[42]-=-, with somewhat different details. We consider the set of all k-element subsets of [n] = f1; 2; \Delta \Delta \Delta ; ng, this is J(n; k). We assume that 2ksn, and look for subsets F ae J(n; k) such ... |

22 | Generalized rook polynomials and orthogonal polynomials, q-Series and Partitions,
- Gessel
- 1989
(Show Context)
Citation Context ...ration and classical orthogonal polynomials. The second aspect of orthogonal polynomials in combinatorics to be discussed is enumeration. There are many ways polynomials appear, including rook theory =-=[23]-=- and matching theory [25]. In this section we consider examples related to the exponential formula. One may consider a set of objects S n , which has a natural decomposition into smaller disjoint sets... |

21 | An A2 Bailey lemma and Rogers–Ramanujan–type identities,
- Andrews, Schilling, et al.
- 1999
(Show Context)
Citation Context ...for what is known) should be found, they presumably lead to a multivariable Askey-Wilson integral, perhaps one due to Gustafson [27]. This may be related to Rogers-Ramanujan identities of higher rank =-=[2]-=-. A combinatorial proof of the Askey-Wilson integral (4.5) exists (see [31]). A combinatorial proof of an equivalent form of (4.4) by counting subspaces of a finite field of order q exists, we review ... |

20 | Laguerre polynomials, weighted derangements and positivity,
- Foata, Zeilberger
- 1988
(Show Context)
Citation Context ...lta \Delta \Delta +n i g is a vertex of type i. This is a theorem of Azor, Gillis, and Victor [3], it was proven using the Viennot machinery in [12]. Other versions exist, for Legendre [26], Laguerre =-=[20]-=-, Meixner [11], Krawtchouk, Charlier, and Meixner-Pollaczek [43] polynomials (conspicuously missing from this list are the ultraspherical polynomials, and Rahman's result for the Jacobi polynomials), ... |

19 |
A combinatorial proof of the Mehler formula”,
- Foata
- 1978
(Show Context)
Citation Context ...rmite polynomial. Foata [17] showed that non-trivial results could result in this way, using combinatorial techniques. One of his most beautiful results, which has been very influential, is his proof =-=[18]-=- of the Mehler formula for Hermite polynomials. It uses the idea of an exponential generating function, which we now explain. We consider graphs on n vertices whose vertices are labeled with the integ... |

14 |
Combinatorial interpretation of integrals of products of Hermite, Laguerre and Tchebycheff polynomials, in:
- Sainte-Catherine, Viennot
- 1984
(Show Context)
Citation Context ... \Delta +n i\Gamma1 +1; \Delta \Delta \Delta ; n 1 + \Delta \Delta \Delta +n i g is a vertex of type i. This is a theorem of Azor, Gillis, and Victor [3], it was proven using the Viennot machinery in =-=[12]-=-. Other versions exist, for Legendre [26], Laguerre [20], Meixner [11], Krawtchouk, Charlier, and Meixner-Pollaczek [43] polynomials (conspicuously missing from this list are the ultraspherical polyno... |

13 |
Combinatorial applications of Hermite polynomials
- Azor, Gillis, et al.
- 1982
(Show Context)
Citation Context ...ta+n k , where any element of fn 1 + \Delta \Delta \Delta +n i\Gamma1 +1; \Delta \Delta \Delta ; n 1 + \Delta \Delta \Delta +n i g is a vertex of type i. This is a theorem of Azor, Gillis, and Victor =-=[3]-=-, it was proven using the Viennot machinery in [12]. Other versions exist, for Legendre [26], Laguerre [20], Meixner [11], Krawtchouk, Charlier, and Meixner-Pollaczek [43] polynomials (conspicuously m... |

13 |
Some q-beta integrals on SU(n) and Sp(n) that generalize the Askey-Wilson and Nassrallah-Rahman integrals
- Gustafson
- 1994
(Show Context)
Citation Context ...complicated. New multilinear versions of the q-Mehler formula (see [30] for what is known) should be found, they presumably lead to a multivariable Askey-Wilson integral, perhaps one due to Gustafson =-=[27]-=-. This may be related to Rogers-Ramanujan identities of higher rank [2]. A combinatorial proof of the Askey-Wilson integral (4.5) exists (see [31]). A combinatorial proof of an equivalent form of (4.4... |

11 |
Linéarisation de produits de polynômes de Meixner, Krawtchouk et Charlier, Publ
- Zeng
- 1988
(Show Context)
Citation Context ...m of Azor, Gillis, and Victor [3], it was proven using the Viennot machinery in [12]. Other versions exist, for Legendre [26], Laguerre [20], Meixner [11], Krawtchouk, Charlier, and Meixner-Pollaczek =-=[43]-=- polynomials (conspicuously missing from this list are the ultraspherical polynomials, and Rahman's result for the Jacobi polynomials), but recently a unified theorem has been given by Kim and Zeng [3... |

10 |
Combinatoire des identites sur les polynomes orthogonaux,
- Foata
- 1983
(Show Context)
Citation Context ...2k fixed points. Here we have p n (x) = bn=2c X k=0 ` n 2k ' \Gamma (2k \Gamma 1)(2k \Gamma 3) \Delta \Delta \Delta 1 \Delta x n\Gamma2k = ~ H n (x); which is a version of a Hermite polynomial. Foata =-=[17]-=- showed that non-trivial results could result in this way, using combinatorial techniques. One of his most beautiful results, which has been very influential, is his proof [18] of the Mehler formula f... |

10 |
A combinatorial approach to the Mehler formulas for Hermite polynomials, Relations between combinatorics and other parts of mathematics
- Foata, Garsia
- 1978
(Show Context)
Citation Context ... a combinatorial study of other classical polynomials: for example Laguerre, Jacobi, Meixner, [6], [11], [34], [35], [40], and to generalized versions of the exponential formula [7]. Foata and Garsia =-=[19]-=- generalized this proof to multilinear generating functions for Hermite polynomials. The Mehler formula is an example of a bilinear generating function, which naturally gives an integral evaluation. F... |

9 | A combinatorial formula for the linearization coefficients of general Sheffer polynomials
- Kim, Zeng
(Show Context)
Citation Context ...3] polynomials (conspicuously missing from this list are the ultraspherical polynomials, and Rahman's result for the Jacobi polynomials), but recently a unified theorem has been given by Kim and Zeng =-=[32]-=-. They consider Sheffer orthogonal polynomials, those that satisfy the recurrence relation (5.2) p n+1 (x) = (x \Gamma (ab + u 3 n + u 4 n))p n (x) \Gamma n(b + n \Gamma 1)u 1 u 2 p n\Gamma1 (x): Note... |

7 | Addition theorems for the q-exponential function., in ”q-series from a contemporary perspective
- Ismail, Stanton
- 1998
(Show Context)
Citation Context ...\Gamma k \Gamma j l q q lj y l : Then three applications of the q-binomial theorem show that (4.7) implies (4.6). Another fundamental generating function whose combinatorics is not understood is (see =-=[29]-=-) (4.8) 1 X n=0 q n 2 =4 H n (xjq) t n (q; q) n = (qt 2 ; q 2 ) 1 E q (x; t); ORTHOGONAL POLYNOMIALS AND COMBINATORICS 13 where E q (x; t) is the quadratic q-exponential function [38], E q (x; t) = (t... |

5 |
Combinatorial proofs of some limit formulas involving orthogonal polynomials
- Labelle, Yeh
- 1990
(Show Context)
Citation Context ...(1 \Gamma t 2 ): This is the third and last term in the exponential of (4.2).sThis proof led to a combinatorial study of other classical polynomials: for example Laguerre, Jacobi, Meixner, [6], [11], =-=[34]-=-, [35], [40], and to generalized versions of the exponential formula [7]. Foata and Garsia [19] generalized this proof to multilinear generating functions for Hermite polynomials. The Mehler formula i... |

5 |
Jacobi polynomials: combinatorics of the basic identities
- Leroux, Strehl
- 1985
(Show Context)
Citation Context ...mma t 2 ): This is the third and last term in the exponential of (4.2).sThis proof led to a combinatorial study of other classical polynomials: for example Laguerre, Jacobi, Meixner, [6], [11], [34], =-=[35]-=-, [40], and to generalized versions of the exponential formula [7]. Foata and Garsia [19] generalized this proof to multilinear generating functions for Hermite polynomials. The Mehler formula is an e... |

4 |
Combinatoire des polynômes orthogonaux classiques: une approche unifiée
- Bergeron
- 1990
(Show Context)
Citation Context ... t 2 y 2 =2(1 \Gamma t 2 ): This is the third and last term in the exponential of (4.2).sThis proof led to a combinatorial study of other classical polynomials: for example Laguerre, Jacobi, Meixner, =-=[6]-=-, [11], [34], [35], [40], and to generalized versions of the exponential formula [7]. Foata and Garsia [19] generalized this proof to multilinear generating functions for Hermite polynomials. The Mehl... |

4 |
A combinatorial interpretation of the integral of the product of Legendre polynomials
- Gillis, Jedwab, et al.
- 1988
(Show Context)
Citation Context ...lta ; n 1 + \Delta \Delta \Delta +n i g is a vertex of type i. This is a theorem of Azor, Gillis, and Victor [3], it was proven using the Viennot machinery in [12]. Other versions exist, for Legendre =-=[26]-=-, Laguerre [20], Meixner [11], Krawtchouk, Charlier, and Meixner-Pollaczek [43] polynomials (conspicuously missing from this list are the ultraspherical polynomials, and Rahman's result for the Jacobi... |

4 |
Un autre q-analogue des nombres d’Euler
- Han, Randrianarivony, et al.
- 1999
(Show Context)
Citation Context ...nd White [41] to new bivariate equidistribution theorems for these two statistics. There are also many relations to continued fractions [15], [16] and determinants, via the moment generating function =-=[28]-=- and Hankel determinants [33]. Because the entries of the determinants are weighted paths, and such determinants are combinatorially understood from the Gessel-Viennot theory, this has led to widespre... |

4 |
Addition theorems for some q-exponential and trigonometric functions
- Suslov
- 1997
(Show Context)
Citation Context ...derstood is (see [29]) (4.8) 1 X n=0 q n 2 =4 H n (xjq) t n (q; q) n = (qt 2 ; q 2 ) 1 E q (x; t); ORTHOGONAL POLYNOMIALS AND COMBINATORICS 13 where E q (x; t) is the quadratic q-exponential function =-=[38]-=-, E q (x; t) = (t 2 ; q 2 ) 1 (qt 2 ; q 2 ) 1 1 X n=0 ` n\Gamma1 Y j=0 (1 + 2iq (1\Gamman)=2+j x \Gamma q 1\Gamman+2j ) ' q n 2 =4 (\Gammait) n (q; q) n ; which satisfies lim q!1 E q (x; t(1 \Gamma q)... |

3 |
The Erdös-Ko-Rado theorem is true for n
- Frankl
- 1976
(Show Context)
Citation Context .... The Erdos-Ko-Rado theorem states that (3.1) jF js` n \Gamma t k \Gamma t ' ORTHOGONAL POLYNOMIALS AND COMBINATORICS 5 as long as nsn 0 (k; t), for n 0 (k; t) which depends only upon k and t. Frankl =-=[21]-=-, and then Wilson [42] found an explicit bound n 0 (k; t) = (k \Gamma t +1)(t+1). It is clear that the set F is realizable by taking all k-subsets which contain a fixed t-element subset. If n ! n 0 (k... |

2 |
An esay proof of the Askey-Wilson integral and applications of the method
- Bowman
(Show Context)
Citation Context ...f h n = 1=jjp n jj 2 , and the bilinear generating function is P (x; y; t) = 1 X n=0 h n p n (x)p n (y)t n ; then (4.3) Z 1 \Gamma1 P (x; y; t)P (x; z; w)d(x) = P (y; z; tw): This was noted by Bowman =-=[8]-=-, who realized that for the q-Hermite polynomials (4.3) becomes the Askey-Wilson integral! The q-Hermite polynomials may be defined by the generating function (q-Hermite GF) 1 X n=0 H n (xjq) t n (q; ... |

2 |
Some formulas for spin models on distance-regular graphs
- Curtin, Nomura
- 1999
(Show Context)
Citation Context ...orem of Delsarte [13] relating the location of zeros of the kernel polynomials for p n () to existence of certain subsets of the graph. Also there is very interesting recent work of Curtin and Nomura =-=[10]-=- classifying spin models motivated from knot theory. Several conjectures are stated there concerning the classification of distance regular graphs with specific orthogonal polynomials. 4. Enumeration ... |

1 |
Algebraic combinatorics---recent topics in association schemes, Sugaku 45
- Bannai
- 1993
(Show Context)
Citation Context ...1) an even cycle with edges alternating red-blue, (2) a path of even length, with edges alternating red-blue, (3) a path of odd length, with edges alternating red-blue, red fixed points at both ends, =-=(4)-=- a path of odd length, with edges alternating red-blue, blue fixed points at both ends. In our example these four possibilities all occur: 3 blue \Gamma\Gamma! 4 red \Gamma\Gamma! 8 blue \Gamma\Gamma!... |

1 |
Medicis, Combinatorics of Meixner polynomials: linearization coefficients
- de
- 1998
(Show Context)
Citation Context ...y 2 =2(1 \Gamma t 2 ): This is the third and last term in the exponential of (4.2).sThis proof led to a combinatorial study of other classical polynomials: for example Laguerre, Jacobi, Meixner, [6], =-=[11]-=-, [34], [35], [40], and to generalized versions of the exponential formula [7]. Foata and Garsia [19] generalized this proof to multilinear generating functions for Hermite polynomials. The Mehler for... |

1 |
Linearisation de produits de polynômes de
- Zeng
- 1990
(Show Context)
Citation Context ... of Azor, Gillis, and Victor [3], and was proven using the Viennot machinery in [12]. Other versions exist, for Legendre [24], Laguerre [18], Meixner [11], Krawtchouk, Charlier, and Meixner-Pollaczek =-=[40]-=- polynomials (conspicuously missing from this list are the ultraspherical polynomials), but recently a unified theorem has been given by Kim and Zeng [30]. They consider Sheffer polynomials, those tha... |