## Appell polynomials and their relatives (2004)

Venue: | Int. Math. Res. Not |

Citations: | 16 - 1 self |

### BibTeX

@ARTICLE{Anshelevich04appellpolynomials,

author = {Michael Anshelevich},

title = {Appell polynomials and their relatives},

journal = {Int. Math. Res. Not},

year = {2004},

pages = {3469--3531}

}

### OpenURL

### Abstract

ABSTRACT. This paper summarizes some known results about Appell polynomials and investigates their various analogs. The primary of these are the free Appell polynomials. In the multivariate case, they can be considered as natural analogs of the Appell polynomials among polynomials in noncommuting variables. They also fit well into the framework of free probability. For the free Appell polynomials, a number of combinatorial and “diagram ” formulas are proven, such as the formulas for their linearization coefficients. An explicit formula for their generating function is obtained. These polynomials are also martingales for free Lévy processes. For more general free Sheffer families, a necessary condition for pseudo-orthogonality is given. Another family investigated are the Kailath-Segall polynomials. These are multivariate polynomials, which share with the Appell polynomials nice combinatorial properties, but are always orthogonal. Their origins lie in the Fock space representations, or in the theory of multiple stochastic integrals. Diagram formulas are proven for these polynomials as well, even in the q-deformed case. 1.

### Citations

381 | Swarttouw,The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
- Koekoek, R
- 1998
(Show Context)
Citation Context ...t of this, it is interesting to look at their q-deformations. For orthogonal polynomials, such deformations are given by the members of the Askey scheme of basic hypergeometric orthogonal polynomials =-=[29]-=-. On the other hand, a possible q-deformed probability theory was initiated by Bo˙zejko and k∏ i=1 xu(i) ] □ □30 M. ANSHELEVICH Speicher [16]. We show that the Kailath-Segall polynomials can be exten... |

190 |
J.A.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials
- Askey, Wilson
- 1985
(Show Context)
Citation Context ... b = 0, the polynomials are the Chebyshev polynomials of the second kind. For general a ∈ R, b ∈ R+, such polynomials have been considered by a number of authors; see the discussion in pages 26–28 of =-=[12]-=-, and also [24]. Definition 6. Let R be a free cumulant generating function, and U(z) an n-tuple of non-commutative power series such that Ui(z) = zi + higher-order terms. Multivariate free Sheffer po... |

148 |
Sur les partitions non croisées d’un cycle
- Kreweras
- 1972
(Show Context)
Citation Context ...lattice of all partitions of the set {1, 2, . . ., n}. For π ∈ P(n), Sing(π) is the collection of single-element (singleton) classes of π. By NC(n) we’ll denote the lattice of non-crossing partitions =-=[31]-=-. These are the partitions with the property that i < j < k, i π ∼ k, j π ∼ l, i π ̸∼ j ⇒ i < l < k.APPELL POLYNOMIALS AND THEIR RELATIVES 5 For a non-crossing partition π, a class B is called outer ... |

143 | Combinatorial aspects of continued fractions
- Flajolet
- 1980
(Show Context)
Citation Context ...ents of the measure can be expressed in terms of each other, and their properties related to the properties of the measure and the orthogonal polynomials, for example using the ViennotFlajolet theory =-=[22, 46, 47]-=-. A typical question in this direction is to find explicitly the linearization coefficients ∫ Pn1(x)Pn2(x) . . .Pnk (x) dµ(x). R Date: March 10, 2008. 2000 Mathematics Subject Classification. Primary ... |

134 | Orthogonal Polynomials of Several Variables - Dunkl, Xu - 2001 |

122 |
Symmetries of some reduced free product C∗- algberas. In Operator algebras and their connections with topology and ergodic theory (Bucsteni
- Voiculescu
- 1983
(Show Context)
Citation Context ...x, y are independent with respect to ϕ if for any P, Q, ϕ [P(x)Q(y)] = ϕ [P(x)]ϕ[Q(y)]. In the early 1980’s, Dan Voiculescu introduced a parallel but really very different notion of free independence =-=[48]-=-. Let ϕ now be a real linear functional on the algebra R〈x, y〉 of polynomials in two non-commuting variables. Then x, y are freely independent with respect to ϕ if whenever ϕ [P1(x)] = ϕ [Q1(y)] = . .... |

86 |
Théorie géométrique des polynômes eulériens
- Foata, Schützenberger
- 1970
(Show Context)
Citation Context ...e one-element cycles of σ. Also, s(σ) is the number of inversions of the permutation ( ) 1 . . . n F(σ) = . u(1, 1) . . . u(k, sk) (F is almost, but not quite, the fundamental transformation of Foata =-=[23]-=-). (c) Let ⃗ui ∈ Ns(i) , i = 1, 2, . . ., k, N = ∑k i=1 s(i), and ⃗u = (⃗u1,⃗u2, . . ., ⃗uk). The product of q-Kailath Segall polynomials can be expanded as k∏ i=1 W ( ) fui(1), fui(2), . . .,fui(s(i)... |

78 |
Une théorie combinatoire des polynômes orthogonaux
- Viennot
- 1984
(Show Context)
Citation Context ...ents of the measure can be expressed in terms of each other, and their properties related to the properties of the measure and the orthogonal polynomials, for example using the ViennotFlajolet theory =-=[22, 46, 47]-=-. A typical question in this direction is to find explicitly the linearization coefficients ∫ Pn1(x)Pn2(x) . . .Pnk (x) dµ(x). R Date: March 10, 2008. 2000 Mathematics Subject Classification. Primary ... |

68 |
An example of a generalized Brownian motion
- Bozejko, Speicher
- 1991
(Show Context)
Citation Context ...key scheme of basic hypergeometric orthogonal polynomials [29]. On the other hand, a possible q-deformed probability theory was initiated by Bo˙zejko and k∏ i=1 xu(i) ] □ □30 M. ANSHELEVICH Speicher =-=[16]-=-. We show that the Kailath-Segall polynomials can be extended to this context, with the same relation to the deformed probability theory and orthogonal polynomials as in the q = 0, 1 cases. As a conse... |

68 | q-Gaussian processes : Non-commutative and classical aspects
- Bożejko, Kummerer, et al.
- 1997
(Show Context)
Citation Context ...q-Brownian motion, and if the process is a q-Poisson process. In all of these cases, from the existence of such martingale polynomials one can deduce the Markov property for the corresponding process =-=[17, 8]-=-. We show that under mild conditions, in all other situations there is no fifth-degree polynomial P(x, t) which is a martingale for such a q-Lévy process. This is a strong indication that such a proce... |

53 |
Recurrence relations, continued fractions, and orthogonal polynomials
- Askey, Ismail
- 1984
(Show Context)
Citation Context ...ials were defined in [2] as all polynomials other than the Meixner families characterized by a certain convolution property. Their measure of orthogonality was found □34 M. ANSHELEVICH explicitly in =-=[11]-=-. The interpretation of these polynomials as q-analogs of the Meixner families was explicitly conjectured in [5], and proved in [3]. We found the following proof independently, and in our particular c... |

45 | Free probability theory and non-crossing partitions, Sém
- Speicher
- 1997
(Show Context)
Citation Context ... with Lemma 9. 3.4. Free binomial. Suppose π ∈ P(n) has the property that the collections {Xi : i ∈ B} B∈π are freely independent. Then the basic relation between free independence and free cumulants =-=[44]-=- says that (27) R(z) = ∑ R(zi : i ∈ B). B∈π More precisely, if R is the free cumulant generating function for {X1, . . .,Xn} (and so a function of z1, . . .,zn), and for each subset B ⊂ {1, . . .,n}, ... |

42 |
Some properties of crossings and partitions
- Biane
- 1997
(Show Context)
Citation Context ...{i ∈ B, i > k}, (S′, π ′) = (S, π) ↾ {k + 1, . . .,j − 1}. Let rc (S, π) = ∑n k=1 rc (k, S, π). Note that also rc (S, π) = rc (π) + ∑ |C ∈ π : min(C) < min(B) < max(C)|, where rc (π) = rc (∅, π) (see =-=[14]-=-). B∈S 2.1.3. Cumulants. A measure µ on R all of whose moments are finite induces a positive semidefinite unital linear functional ϕ on R[x] by ϕ [x n ] = mn(µ). Positivity will not play a part in mos... |

42 |
Orthogonale polynomsysteme mit einer besonderern der erzeugenden funktion
- Meixner
- 1934
(Show Context)
Citation Context ...the multinomial expansion properties and the relation to stochastic processes (see Section 2.4). Among the Appell polynomials, only the Hermite ones are orthogonal. Meixner’s classic characterization =-=[34]-=- describes all the orthogonal Sheffer polynomials. There are also multivariate versions of this statement [38]. We start the paper by describing some properties of and relations between the three afor... |

41 | On the multiplication of free N-tuples of noncommutative random variables
- Nica, Speicher
- 1996
(Show Context)
Citation Context ...ICH corresponding free Lévy process. This is the case if ψ itself is freely infinitely divisible. However, in the free one-dimensional case, more is true: for any positive ψ, ψt is positive for t ≥ 1 =-=[35]-=-. 2.4. Martingales. Let {X(t)} t∈[0,∞) be a Lévy process, that is, a stochastic process with stationary independent increments. Assume that all of the joint moments of {X(t)} (with respect to the expe... |

38 | Stochastic Processes and Orthogonal Polynomials - Schoutens - 2000 |

20 |
Diagonalization of certain integral operators
- Ismail, Zhang
- 1994
(Show Context)
Citation Context ... we consider a q-deformation of the Meixner families. One such deformation has been considered in [27]. It is based on the exponential function for the operator calculus for the Askey-Wilson operator =-=[26]-=-. Previously, Al-Salam showed that under this approach, the unique orthogonal Appell family also consists of (multiples of) Rogers q-Hermite polynomials [4]. Under the more elementary approach of [1],... |

18 |
Partition-dependent stochastic measures and q-deformed cumulants
- ANSHELEVICH
- 2001
(Show Context)
Citation Context ...]q! Pn(x, t)z n ∞∏ 1 = H(x, t, z) = 1 − (1 − q)xU(zq k ) + R(U(zqk )) − R(U(zqk+1 )) . n=0 k=0 4.4. q-Kailath-Segall polynomials. The origin of these polynomials is in the q-Lévy processes defined in =-=[6, 9]-=-. Definition 8. Let A0 be a complex star-algebra without identity, and 〈·〉 a star-linear functional on it. Let A be the complex unital star-algebra generated by non-commuting symbols {X(f) : f ∈ Asa 0... |

16 |
Convolutions of orthonormal polynomials
- Al-Salam, Chihara
- 1976
(Show Context)
Citation Context ...ls defined in [41] are, up to a shift, of this form, with a = 1 − 2p, b = −p(1 − p), 2t = Np(1 − p), except that in this case, as expected, b is negative. Al-Salam-Chihara polynomials were defined in =-=[2]-=- as all polynomials other than the Meixner families characterized by a certain convolution property. Their measure of orthogonality was found □34 M. ANSHELEVICH explicitly in [11]. The interpretation... |

11 | Feynman diagrams and Wick products associated with q-Fock space
- Effros, Popa
- 2003
(Show Context)
Citation Context ... words, i(⃗u) is the number of inversions of the permutation ( ) 1 . . . k k + 1 . . . n . u(1) . . . u(k) 1 . . . nAPPELL POLYNOMIALS AND THEIR RELATIVES 37 Many of the following formulas appear in =-=[21]-=- in the q-Gaussian case A0 = C0〈x〉 and 〈x⃗u〉 = δ|⃗u|,2. Theorem 27. The following expansions hold. (a) A monomial in X(f)’s can be expanded in terms of the q-Kailath-Segall polynomials: X(f1)X(f2) . .... |

11 |
Multivariate Appell polynomials and the central limit theorem
- Giraitis, Surgailis
- 1986
(Show Context)
Citation Context ...ulus [40] and the study of hypergroups, in numerical analysis (Bernoulli polynomials are Appell), but also in probability theory, in the study of stochastic processes [32], non-central limit theorems =-=[13, 25]-=-, and natural exponential families [39]. From the combinatorial point of view, they have nice linearization and multinomial formulas. The third family of polynomials has not apparently been explicitly... |

11 | Bernadette Combinatorics of free cumulants
- Speicher, Krawczyk
(Show Context)
Citation Context ... the lattice of all partitions. The following proof is also very similar to the one in [25]. Proof. Equation (29) is the basic relation between moments and free cumulants. Equation (31) was proven in =-=[30, 45]-=-. We prove equation (32); the proof of (30) is similar, and also implied by equation (34) below. In fact, we will prove a more general statement, that (33) R [ A (X⃗u1) , . . .,A ( ) X⃗uj , (X⃗uj+1 ),... |

8 |
Orthogonal functionals of independent-increment processes
- Segall, Kailath
- 1976
(Show Context)
Citation Context ...al point of view, they have nice linearization and multinomial formulas. The third family of polynomials has not apparently been explicitly defined before, although it appears implicitly in the paper =-=[43]-=-. For this reason, we will call them the Kailath-Segall polynomials. These are polynomials in (infinitely many) variables {xk} ∞ k=1 . They are indexed by all finite sequences of natural numbers ⃗u = ... |

7 |
A q-deformed Poisson distribution based on orthogonal polynomials
- Saitoh, Yoshida
(Show Context)
Citation Context ...ontinuous) q-Hermite polynomials, while the Charlier case b = 0 corresponds to what are usually called the continuous big q-Hermite polynomials. Note also that the q-Krawtchouk polynomials defined in =-=[41]-=- are, up to a shift, of this form, with a = 1 − 2p, b = −p(1 − p), 2t = Np(1 − p), except that in this case, as expected, b is negative. Al-Salam-Chihara polynomials were defined in [2] as all polynom... |

6 |
q-Pollaczek polynomials and a conjecture of Andrews and
- Al-Salam, Chihara
- 1987
(Show Context)
Citation Context ...easure of orthogonality was found □34 M. ANSHELEVICH explicitly in [11]. The interpretation of these polynomials as q-analogs of the Meixner families was explicitly conjectured in [5], and proved in =-=[3]-=-. We found the following proof independently, and in our particular case it is also somewhat simpler. For a more interesting characterization using stochastic processes, see [18]. Theorem 24. Suppose ... |

6 |
Sur une classe de polynômes
- Appell
(Show Context)
Citation Context ..., 2008. 2000 Mathematics Subject Classification. Primary 05A; Secondary 46L54. This work was supported in part by an NSF postdoctoral fellowship. 12 M. ANSHELEVICH Another natural and very classical =-=[9]-=- polynomial family associated to µ is its family of Appell polynomials, which have the exponential generating function ∞∑ 1 (1) n! A(n) (x)z n = exp(xz − log Mµ(z)), where n=0 Mµ(z) = ∞∑ n=0 1 n mn(µ)... |

6 | A combinatorial formula for the linearization coefficients of general Sheffer polynomials
- Kim, Zeng
(Show Context)
Citation Context ... in part by an NSF postdoctoral fellowship. 1 R2 M. ANSHELEVICH Already the proofs of the positivity of these coefficients are quite subtle [19] and they are known explicitly only in very rare cases =-=[28]-=-. Another natural and very classical [10] polynomial family associated to µ is its family of Appell polynomials, which have the exponential generating function ∞∑ 1 (1) n! A(n) (x)z n = exp(xz − log M... |

4 |
martingale polynomials
- Free
(Show Context)
Citation Context ... pseudo-orthogonal polynomials. 3.6. Free Sheffer and Meixner families. Sheffer families are martingale polynomials for the corresponding Lévy processes. Based on this idea and the result of [15], in =-=[7]-=- we proposed the definition of free Sheffer families, which are families of martingale polynomials with respect to free Lévy processes. Specifically, free Sheffer families are families of monic polyno... |

4 | Combinatorial orthogonal expansions
- Médicis, Stanton
- 1996
(Show Context)
Citation Context ...tion. Primary 05A; Secondary 46L54. This work was supported in part by an NSF postdoctoral fellowship. 1 R2 M. ANSHELEVICH Already the proofs of the positivity of these coefficients are quite subtle =-=[19]-=- and they are known explicitly only in very rare cases [28]. Another natural and very classical [10] polynomial family associated to µ is its family of Appell polynomials, which have the exponential g... |

3 |
Classical orthogonal polynomials, Orthogonal polynomials and applications (Bar-le-Duc
- Andrews, Askey
- 1984
(Show Context)
Citation Context ...n property. Their measure of orthogonality was found □34 M. ANSHELEVICH explicitly in [11]. The interpretation of these polynomials as q-analogs of the Meixner families was explicitly conjectured in =-=[5]-=-, and proved in [3]. We found the following proof independently, and in our particular case it is also somewhat simpler. For a more interesting characterization using stochastic processes, see [18]. T... |

3 |
Noncentral limit theorems and Appell
- Avram, Taqqu
- 1987
(Show Context)
Citation Context ...ulus [40] and the study of hypergroups, in numerical analysis (Bernoulli polynomials are Appell), but also in probability theory, in the study of stochastic processes [32], non-central limit theorems =-=[13, 25]-=-, and natural exponential families [39]. From the combinatorial point of view, they have nice linearization and multinomial formulas. The third family of polynomials has not apparently been explicitly... |

3 |
with free increments
- Processes
- 1998
(Show Context)
Citation Context ... ANSHELEVICH 3.5. Martingale property. The martingale property of free Appell (and, more generally, Sheffer) polynomials for processes with freely independent increments was shown previously by Biane =-=[15]-=-. The following is an alternative proof for distributions all of whose moments are finite, which uses the binomial property above. Lemma 15. Let {µt} be a free convolution semigroup with all moments f... |

3 | On the computation of spectra in free probability
- Lehner
(Show Context)
Citation Context ...e, A (X⃗u) = k∏ i=1 ⃗u A (X⃗vi ) . The proof is based on the following lemma due to Franz Lehner. The lemma deals with bounded operators, but it applies equally well to formal power series. Lemma 14. =-=[33]-=- Let S1, . . .,SN ∈ B(H) be arbitrary operators and assume that the sum of alternating products ∞∑ ∑ I + Si1Si2 . . .Sin n=1 i1̸=i2̸=...̸=in (the sum over all products where neighboring factors are di... |

3 |
on free probability theory, Lectures on probability theory and statistics (Saint-Flour
- Lectures
- 1998
(Show Context)
Citation Context ...ever ϕ [P1(x)] = ϕ [Q1(y)] = . . . = ϕ [Pn(x)] = ϕ [Qn(y)] = 0 and Q0, Pn+1 each are either centered or scalar, then ϕ [Q0(y)P1(x)Q1(y) . . .Pn(x)Qn(y)Pn+1(x)] = 0. A whole theory of free probability =-=[49]-=-, based on this notion, is by now quite well developed. It turns out that there are “free analogs of” the Appell and Kailath-Segall polynomials, which, very4 M. ANSHELEVICH roughly, are obtained by r... |

2 |
A strategy for determining polynomial orthogonality, Mathematical essays in honor of Gian-Carlo Rota
- Freeman
- 1996
(Show Context)
Citation Context ...ynomials are the Chebyshev polynomials of the second kind. For general a ∈ R, b ∈ R+, such polynomials have been considered by a number of authors; see the discussion in pages 26–28 of [12], and also =-=[24]-=-. Definition 6. Let R be a free cumulant generating function, and U(z) an n-tuple of non-commutative power series such that Ui(z) = zi + higher-order terms. Multivariate free Sheffer polynomials are d... |

2 |
Martingales and boundary crossing probabilities for Markov processes, Ann. Probability 2
- Lai
- 1974
(Show Context)
Citation Context ...hey arise in finite operator calculus [40] and the study of hypergroups, in numerical analysis (Bernoulli polynomials are Appell), but also in probability theory, in the study of stochastic processes =-=[32]-=-, non-central limit theorems [13, 25], and natural exponential families [39]. From the combinatorial point of view, they have nice linearization and multinomial formulas. The third family of polynomia... |

2 |
R-transforms in free probability, lecture notes for an IHP course (unpublished
- Nica
(Show Context)
Citation Context ... can be summarized in a relation between their generating functions, as follows. The following proposition is due to Nica and Speicher; for completeness, we provide a direct proof. ( ) Proposition 6. =-=[36]-=- Let zi = wi 1 + M(w) . Then ( ( ) ( ) R(z) = R w1 1 + M(w) , . . ., wn 1 + M(w) ) = M(w). Proof. In the defining relation (19), the sum is over all non-crossing partition. Each non-crossing partition... |

2 |
Nualart and Wim Schoutens, Chaotic and predictable representations for Lévy processes, Stochastic Process
- David
(Show Context)
Citation Context ... ∗ ⊥ W⃗v (x) = 0. Free and even q-analogs of Corollary 4 hold, derived from appropriate modifications of Corollary 2. These properties are related to the “generalized chaos decomposition property” of =-=[37]-=-. Appell polynomials are linear combinations of the Kailath-Segall polynomials of the same degree. Note that a priori, such a linear combination is a multivariate polynomial, but in this case it turns... |

2 |
Orthogonality of the Sheffer system associated to a Levy process
- Pommeret
(Show Context)
Citation Context ...ell polynomials, only the Hermite ones are orthogonal. Meixner’s classic characterization [34] describes all the orthogonal Sheffer polynomials. There are also multivariate versions of this statement =-=[38]-=-. We start the paper by describing some properties of and relations between the three aforementioned families of polynomials. In the Appell and Kailath-Segall cases, natural starting points are in fac... |

1 |
Al-Salam, A characterization of the Rogers q-Hermite polynomials
- Waleed
- 1995
(Show Context)
Citation Context ...olynomials. There is a number of different possible definitions for q-Appell polynomials. For example, two q-deformations of the relation ∂xAn = nAn−1, leading to such definitions, were considered in =-=[1, 4]-=-. We prefer, instead, to use an interpolation between the recursion relations in the classical and the free cases. Definition 7. The q-Appell polynomials are defined via the recursion relation n∑ ( ) ... |

1 |
An operator calculus for the Askey-Wilson operator, Ann
- Ismail
- 1999
(Show Context)
Citation Context ...[n − k]q![k] (√ 1 − q) k−1 Pk(x)Un−k−1( √ 1 − q x). 4.3. Al-Salam-Chihara polynomials. In this section we consider a q-deformation of the Meixner families. One such deformation has been considered in =-=[27]-=-. It is based on the exponential function for the operator calculus for the Askey-Wilson operator [26]. Previously, Al-Salam showed that under this approach, the unique orthogonal Appell family also c... |

1 |
Orthogonal polynomsysteme mit einer besonderen gestalt der erzengenden
- Meinxer
- 1934
(Show Context)
Citation Context ...the multinomial expansion properties and the relation to stochastic processes (see Section 2.4). Among the Appell polynomials, only the Hermite ones are orthogonal. Meixner’s classic characterization =-=[34]-=- describes all the orthogonal Sheffer polynomials. There are also multivariate versions of this statement [38]. We start the paper by describing some properties of and relations between the three afor... |

1 |
and pseudo-orthogonal multi-dimensional Appell polynomials
- Orthogonal
(Show Context)
Citation Context ...numerical analysis (Bernoulli polynomials are Appell), but also in probability theory, in the study of stochastic processes [32], non-central limit theorems [13, 25], and natural exponential families =-=[39]-=-. From the combinatorial point of view, they have nice linearization and multinomial formulas. The third family of polynomials has not apparently been explicitly defined before, although it appears im... |