Linearization coefficients for orthogonal polynomials Michael Anshelevich February 26, 2003
P n = monic polynomials of degree n = 0, 1,.... {P n } = basis for the polynomials in 1 variable. Linearization coefficients: coefficients in the expansion of in the basis {P n }. P n1 P n2... P nk For many classical families of orthogonal polynomials, these are positive. For some examples, in fact positive integers. 1
Let µ be a positive measure on R such that its moments m n (µ) = R xn dµ(x) <. {P n } n=0 = Gram-Schmidt orthogonalization of {xn } n=0. These are, up to normalization, the orthogonal polynomials for µ, P n (x)p k (x)dµ(x) = P n P k = 0 if n k. Theorem (Favard s theorem). {P n } n=0 are the monic orthogonal polynomials for some positive functional iff they satisfy a three-term recursion relation xp n = P n+1 + α n+1 P n + β n P n 1, where P 1 = 0, α n R, β n 0. 2
When {P n } are orthogonal, P n1 P n2... P nk = m=0 1 P 2 m P n1 P n2... P nk P m Pm, where is the expectation with respect to the orthogonality measure, F = F (x)dµ(x). Thus their linearization coefficients are, up to normalization, Pn1 P n2... P nk. Generalizations of moments x n if P 1 (x) = x. 3
Hermite polynomials: xh n (x, t) = H n+1 (x, t) + nth n 1 (x, t). Orthogonal with respect to the Gaussian distribution Moments m n (µ t ) = dµ t (x) = 1 2πt e x2 /2t dx. Linearization coefficients k j=1 H nk (x, t) 0, n odd, t n/2 (n 1)!!, n even = t n/2 {Pair set partitions of n}. = t n/2 P 2 (n 1, n 2,..., n k ). Here P 2 (n 1, n 2,..., n k ) = inhomogeneous pair partitions. 4
Centered Charlier polynomials: xc n (x, t) = C n+1 (x, t) + nc n (x, t) + tnc n 1 (x, t). Orthogonal with respect to the Poisson distribution dµ t (x) = e t j=0 t j j! δ j(x) shifted by t. Moments: for t = 1, related to Bell numbers. More generally, m n (µ t ) = π P(n), s(π)=0 t π. Here the sum is over all set partitions with no singletons, and π = number of classes of the partition. Linearization coefficients k (Zeng 90) j=1 C nk (x, t) = π P(n 1,n 2,...,n k ), s(π)=0 t π. 5
Centered Laguerre polynomials: xl n (x, t) = L n+1 (x, t) + 2nL n (x, t)+ n(t + n 1)L n 1 (x, t). Orthogonal with respect to the Gamma distribution dµ t (x) = 1 Γ(t) xt 1 e x dx shifted by t. Moments: for t = 1, n!. Suggests statistics over permutations. General moments: m n (µ t ) = σ D(n) t cyc(σ). Here cyc (σ) = the number of cycles of σ, D(n) = derangements. Linearization coefficients k j=1 L nk (x, t) (Foata-Zeilberger 88) = σ D(n 1,n 2,...,n k ) t cyc(σ). Here D(n 1, n 2,..., n k ) = inhomogeneous permutations = generalized derangements. 6
Meixner and Meixner-Pollaczek polynomials: xp n (x, t) = P n+1 (x, t) + (α + β)np n (x, t) + n(t + αβ(n 1))P n 1 (x, t), depending on whether α, β are real or β = ᾱ. Appell polynomials: exponential generating function n=0 1 n! P n(x, t)z n = f(z) t e xz. The only orthogonal ones are the Hermite polynomials. Sheffer polynomials: exponential generating function n=0 1 n! P n(x, t)z n = f(z) t e xu(z). The only orthogonal ones are precisely the classes above. 7
Linearization coefficients: k j=1 = P nk (x, t) σ D(n 1,n 2,...,n k ) (Zeng 90, Kim-Zeng 01) α dec(σ) cyc(σ) β exc(σ) cyc(σ) t cyc(σ). Here dec(σ), resp. exc(σ) are the numbers of the descents, resp. ascents in the permutation σ, i.e. the number of i such that σ(i) > i, resp. σ(i) < i. Note: recover Laguerre for α = β = 1, Charlier for α = 1, β = 0, Hermite for α = β = 0. 8
Chebyshev polynomials of the second kind: xu n (x, t) = U n+1 (x, t) + tu n 1 (x, t). Orthogonal with respect to dµ t (x) = 1 4t x 2 1 2πt [ 2 t,2 t] (x)dx. Moments m n (µ t ) = 0, n odd, t n/2 n th Catalan number, n even = t n/2 {Non-crossing pair set partitions of n}. Linearization coefficients k j=1 H nk (x, t) ( de Sainte-Catherine, Viennot 85) = t n/2 NC 2 (n 1, n 2,..., n k ). Here NC 2 (n 1, n 2,..., n k ) = inhomogeneous non-crossing pair partitions. 9
Expect to have k j=1 C 0,nk (x, t) = π NC (n 1,n 2,...,n k ) s(π)=0 t π for some polynomials. Indeed have this for the free Charlier polynomials xc 0,0 (x, t) = C 0,1 (x, t) (+1), xc 0,m (x, t) = C 0,m+1 (x, t) + C 0,m (x, t) + tc 0,m 1 (x, t). These are orthogonal with respect to dµ t (x) = 1 2πx 4t (x t 1) 2 1 [c1,c 2 ] (x)dx + max(1 t, 0)δ 0 (x), where c 1 = t + 1 2 t, c 2 = t + 1 + 2 t. 10
Continuous (Rogers) q-hermite polynomials: xh q,n (x, t) = H q,n+1 (x, t) + t[n] q H q,n 1 (x, t). Here q is (say) in ( 1, 1), and [n] q = n 1 j=0 Orthogonal with respect to q j = 1 qn 1 q. dµ t,q (x) = 1 π (qe 1 q sin(θ)(q; q) 2iθ 2 ; q) dx, t for x = 2 1 q t cos(θ), θ [0, π], and (a; q) = j=0 (1 aq j ). This is a probability measure supported on the interval [ 2 t/ 1 q, 2 t/ 1 q]. 11
Moments m n (µ t,q ) = π P 2 (n) q rc(π). Here rc (π) is the number of crossings of the pair partition π. Linearization coefficients k H nk (x, t) = j=1 π P 2 (n 1,n 2,...,n k ) (Ismail, Stanton, Viennot 87) q rc(π). 12
Centered continuous big q-hermite polynomials, which in our context are q-analogs of the Charlier polynomials. Recursion relations xc q,n (x, t) = C q,n+1 (x, t) + [n] q C q,n (x, t) + t[n] q C q,n 1 (x, t). Moments m n (µ t,q ) = π P(n) s(π)=0 q rc(π) t π, where rc (π) = number of restricted crossings of the partition π. Linearization coefficients k j=1 C q,nk (x, t) = π P(n 1,n 2,...,n k ) s(π)=0 q rc(π) t π. 13
Hermite Charlier Laguerre / Meixner f(z) t e xu(z) q-hermite big q-hermite Al-Salam-Chihara? F (z) 1 k=0 1 u(q k z)x? 1 1+tf(z) u(z)x Chebyshev free Charlier free Meixner 14
PROCESSES ON A q-deformed FULL FOCK SPACE Consider the Hilbert space H = L 2 (R +, dx). Let F alg (H) = H k k=0 be its algebraic Fock space. Here the 0 th component is spanned by the vacuum vector Ω. f 1... f k, g 1... g n 0 = δ kn f 1, g 1... f k, g k is an inner product. Let P q (f 1... f n ) = σ Sym(n) q i(σ) f σ(1)... f σ(n), where Sym(n) is the permutation group and i(σ) is the number of inversions of σ. (Bożejko, Speicher 91) P q 0 for 1 < q < 1. 15
ξ, η q = ξ, P q η 0. another inner product, F q (L 2 (R + )) = the completion of F alg (L 2 (R + ) with respect to the corresponding norm, the q-deformed full Fock space. On F q (L 2 (R + )), define creation, annihilation, and gauge operators a (t), a(t), p(t). The non-commutative stochastic process X(t) = a (t) + a(t) is the q-brownian motion, and the process X(t) = a (t) + a(t) + p(t) is the centered q-poisson process. 16
Corresponding distribution: Ω, f(x(t))ω = f(x)dµ t (x). For the degenerate case q = 1, get the corresponding classical processes. Moments expressed in terms of generalized cumulants: m n = π P(n) R π. Proposition. For the processes on the q-fock space, R π = q rc(π) r B. B π For the Brownian motion, r 2 = t, r n = 0 for n > 2. For the centered Poisson process, r n = t for n > 1. The formulas for the moments follow. 17
COMBINATORIAL STOCHASTIC MEASURES (Rota, Wallstrom 97) {X(t)} = operator-valued stochastic process, stationary wrt, independent increments. For a set partition π = (B 1, B 2,..., B l ), temporarily denote by c(i) the number of the class B c(i) to which i belongs. 18
Stochastic measure corresponding to the partition π is St π (t) = [0,t) l all s i s distinct dx(s c(1) )dx(s c(2) ) dx(s c(n) ). n = Stˆ1 ψ n = Stˆ0 ψ n (t) = the higher diagonal measures n (t) = [0,t) (dx(s))n, the full stochastic measures [0,t) n all s i s distinct dx(s 1 )dx(s 2 ) dx(s n ). 19
t X(t) an operator-valued measure. So imitate Lebesgue integration. More precisely, 1 [u,v) (t)dx(t) = X(v) X(u) and approximate the integrand by simple functions. Convergence shown under various conditions. Proposition (Linearization Itô formula). k j=1 ψ nk = π P(n 1,n 2,...,n k ) St π. Analogy: ψ n P n. In our examples ψ n (t) = P n (X(t), t). Proposition. R π = St π and r n = n. 20
k j=1 ψ nk = π P(n 1,n 2,...,n k ) R π. (1) For a centered process, r 1 = 0. ψ n ψ k = 0 for n k. Thus stochastic measures are analogs of orthogonal polynomials, and formula (1) describes their linearization coefficients. 21
In particular, for the stochastic measures on the q-fock space k j=1 ψ nk = = R π π P(n 1,n 2,...,n k ) q rc(π) π P(n 1,n 2,...,n k ) B π r B. For the q-brownian motion, ψ m (t) = H q,m (X(t), t), a scaled version of the continuous (Rogers) q-hermite polynomials. For the centered q-poisson process, ψ m (t) = C q,m (X(t), t), a scaled version of the centered continuous big q-hermite polynomials. The formulas for the linearization coefficients follow. 22
In fact, in the q-brownian motion case, for π P 1,2, St π = q rc(π)+sd(π) t s 2(π) H q,s(π) (X(t), t) and they are 0 otherwise. Here singleton depth sd (π) = sum of depths, d(i) = { j a, b Bj : a < i < b }, over all the singletons (i) of π. Thus and x n = π P 1,2 (n) H q,n (x, t) = π P 1,2 (n) q rc(π)+sd(π) t s2(π) H q,s1 (π)(x, t) ( 1) s 2(π) q rc(π)+sd(π) t s 2(π) x s 1(π). 23