On generalized Kesten--McKay distributions

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Jan 7, 2016 - arXiv:1507.03191v2 [math.PR] 7 Jan 2016. ON GENERALIZED KESTEN–MCKAY DISTRIBUTIONS. PAWE L J. SZAB LOWSKI. Abstract.
arXiv:1507.03191v2 [math.PR] 7 Jan 2016

ON GENERALIZED KESTEN–MCKAY DISTRIBUTIONS PAWEL J. SZABLOWSKI Abstract. We√ examine properties of distributions with the density of the 2An c2 −x2 form: πc Qn (c(1+a 2 )−2a x) , where c, a1 , . . . , an are some parameters and An j=0

j

j

suitable constant. We find general forms of An , of k−th moment and of k−th polynomial orthogonal with respect to this measure. On the way we prove several identities concerning rational symmetric functions. Finally we consider the case of parameters a1 , . . . , an forming conjugate pairs and give multivariate interpretation of obtained distributions at least for the cases n = 2, 4, 6.

1. Introduction The purpose of this note is to analyze the properties of the following family of distributions having densities of the form: √ 2A c2 − x2 Qn n , (1.1) fKMKn (x|c, a1 , . . . , an ) = πc j=1 (c(1 + a2j ) − 2aj x)

defined for n ≥ 0, |x| ≤ c, with c > 0, |aj | < 1, j = 1, . . . , n. Here An is a normalizing constant being the function of parameters c, a1 , . . . , an . We will call this family generalized Kesten–McKay distributions. The name is justified by the fact that the distribution with the following density p v 4(v − 1) − x2 2 (1.2) fKMK2 (x|4/a , a, −a) = , 2π(v 2 − x2 )

where v = 1+1/a2 and |a| < 1 has been defined, described and what is more, derived in [4] or [8]. Then the name Kesten-McKay distribution has been attributed to this distribution in the literature that appeared after 1981. Thus it is justified to call the distribution defined by (1.1) a generalized KestenMcKay (GKM) distribution. Note also that for n = 0 fKMK0 (x|c) becomes Wigner or semicircle distribution with parameter c. It should be underlined that for n = 0, 1, 2 distributions of this kind appear not only in the context of random matrices, random graphs which is typical application of Kesten–Mckay distributions (see e.g.[6], [7], [9]) but also in the context of the so-called free probability a part of non-commutative probability theory recently rapidly developing. One of the first papers where semicircle and related distribution appeared in the non-commutative probability context is [1]. Date: June 2015. 2010 Mathematics Subject Classification. Primary 60E05,05E05; Secondary 62H05,60J10. Key words and phrases. Kesten–McKay distributions, Chebyshev polynomials, orthogonal polynomials, moments, symmetric rational functions, multivariate distributions. 1

2

PAWEL J. SZABLOWSKI

For n < 5 distribution fKMKn can be identified as the special case of the Askey– Wilson chain of distribution that make 5 families of polynomials of the so-called Askey–Wilson scheme of polynomials, orthogonal. For the reference see e.g. [12] For n ≥ 5 the distributions fKMKn were not yet described. We will present unified approach and recall and collect information on this family that is scattered though literature. Let us observe first that if X ∼ fKMKn (x|c, a1 , . . . , an ) and Y ∼ fKMKn (x|1, a1 , . . . , an ) then X ∼ cY. Hence we will consider further only distributions with the density fKMKn (x|1, a1 , . . . , an ) which we will denote for brevity fKn (x|a1 , . . . , an ). To start analysis let us recall that 1/(1 + a2j − 2aj x) =

(1.3)

∞ X

akj Uk (x),

k=0

where Uk denotes k−th Chebyshev  polynomial of the second kind and that √ R1 2 0 if j 6= k 2 . Hence the form of the density −1 π Uk (x)Uj (x) 1 − x dx = 1 if j = k (1.1) fits the scheme of distributions and orthogonal polynomials that was considered in [12] and hence we can use ideas and results presented there. First of all let us observe that the densities considered in this paper are the special cases of the distributions known from the so-called Askey–Wilson scheme of distributions and orthogonal polynomials obtained from the general ones by setting q = 0. q is a special parameter called base in the theory of Askey–Wilson polynomials. More precisely for n = 0 we deal with Wigner distribution, for n = 1 we deal with the so-called continuous big q−Hermite polynomials and the distribution that makes them orthogonal when we set q = 0, for n = 2 we deal with the so-called Al-Salam– Chihara polynomials and the distribution that make these polynomials orthogonal (of course for q = 0), for n = 3 we deal with the so-called dual Hahn polynomials and the distribution that make these polynomials orthogonal. Finally for n = 4 we deal with the so-called Askey–Wilson polynomials and the distribution that make them orthogonal. Hence in particular we know the families of orthogonal polynomials that our distributions for n = 0, . . . , 4 make orthogonal. For the precise definitions and further references see e.g. [5] or [12]. In the sequel we will use exchangeable an and (a1 , . . . , an ) depending on the required brevity and Qn clarity. Qn Let us denote j=1,j6=i as j6=i . We have the following general observation. Theorem 1. If ai 6= aj , i 6= j, i, j = 1, . . . , n, then i) constant An in (1.1) is given by: (1.4)

An = An (an ) = 1/

n X i=1

ain−1 , j6=i (ai − aj )(1 − ai aj )

Qn

ii) besides we have (1.5) Bn,k = Bn,k (an ) =

Z

1

−1

Uk (x)fKn (x|an )dx = An

n X i=1

an+k−1 i . j6=i (ai − aj )(1 − ai aj )

Qn

KESTEN–MCKAY

3

Proof. We will use the fact that 1/

n Y

(x − bi ) =

i−1

n X i=1

1 1 , (x − bi ) (b − b ) i j j=1,j6=i

Qn

and the fact that ((1 + a2 )/a − (1 + b2 )/b) = (b − a)(1 − ab)/(ab). Hence we have n Y 2An (−1)n p 2/ Q (x − (1 + a2i )/(2ai )) 1 − x n 2n π i=1 ai i=1 Qn n X ain−1 j6=i aj 1 2An (−1)n p Qn = n Qn 1 − x2 2 2 π i=1 ai j6=i (aj − ai )(1 − ai aj ) (x − (1 + ai )/(2ai )) i=1

fKn (x|a1 , . . . , an ) =

n

=

X 1 2An (−1)n p ain−1 Qn 1 − x2 2 ) − 2a x) . (a − a )(1 − a a ) π ((1 + a j i j i i j6=i i i=1

Now following (1.3) and orthogonality of polynomials U ′ s with respect to Wigner masure we have: √ Z 1 2 1 − x2 Uk (x)dx = aki . 2 −1 π((1 + ai ) − 2ai x)  Remark 1. Since coefficients Bn,k are coefficients in the orthogonal expansion in L2 ([−1, 1], w), with basis {Uk }k≥0 . we get the following expansion for free: ∞

fKn (x|an ) =

X 2p 1 − x2 Bn,k Uk (x). π k=0

Above w denoted Wigner measure with the density of such expansions see [10].

2 π

√ 1 − x2 . For more examples

In the sequel we will need the following quantities: (1.6)

(n)

Sk (an ) =

X

k Y

ajm

0≤j1