[Math.CO] 3 Jul 2002

arXiv:math/0110031v2 [math.CO] 3 Jul 2002 fcneuiepit scle a called is points consecutive of osbesteps possible nters fti ae l atc ah ilhv h olwn prope following the have will paths lattice all paper this of rest the In ento 2.1. Definition eue oepesteJcb aaeesi em fcumulants. of terms in parameters Jacobi the express to used be parameters. Jacobi gav coefficients he the for Lascoux, 35]. formula [16, A. general cf. with more fractions, discussions a of After functions paths. symmetric lattice of theory 3]. 26, behavio 6, been the have [22, proje generators, understand e.g. free free see of of to case case interesting the special to equivalent be The translation would convolution. developm free it the under and for polynomials movitations [29] Voiculescu’s theory of Spectra ity one [2]. also theory was spectral [28]. operators and survey problems the moment synth in with A connection or [27] polynomials. memoir orthogonal Viennot’s of in found theory for be the [1] frac h can of and continued principle walks ingredient times random basic basic Stieltjes’ to The a Since application survey. an probability. for noncommutative a o [11] in for and e.g. McGregor [14] see and or times, Karlin [10] many of see work enumeratio to processes, for connected death is expressions This function paths. generating lattice derived and tions otne rcin,Hne eemnns rhgnlpolynom orthogonal determinants, Hankel fractions, continued ial nscin6w niaehwGse-ino hoyo aklde Hankel of theory Gessel-Viennot how cum indicate free we expressing 6 section formula in a Finally result, main cumulants. the various prove of we orthogona definitions 5 about the section facts review In relevant we function the 4 generating discuss section the briefly In for we formula 3 Flajolet’s section review In we 2 follows. section as In organized is paper The Flajole spirit the in cumulants free for formula a derive we note this In ext been have operators Jacobi their and polynomials Orthogonal interpretation combinatorial a gave Flajolet [9] paper seminal his In e od n phrases. and words Key 1991 Date coe 6 2018. 26, October : ahmtc ujc Classification. Subject Mathematics ino hoyt xrs akldtriat ntrso variou of terms in determinants conv Hankel the express For to formula. Flajo theory inversion combines Viennot Lagrange It the and derived. fractions tinued is polynomials orthogonal corresponding Abstract. UUAT,LTIEPTS N ORTHOGONAL AND PATHS, LATTICE CUMULANTS, Z 2 × A oml xrsigfe uuat ntrso aoiparameter Jacobi of terms in cumulants free expressing formula A Z atc path lattice 2 auto fapt stepouto h autoso t steps. its of valuations the of product the is path a of valuation A . uuat,ltiept obntrc,Mtknpts Lukasiewic paths, Motzkin combinatorics, path lattice Cumulants, 2. nmrto fLtiePaths Lattice of Enumeration sasqec fpit nteitgrlattice integer the in points of sequence a is step POLYNOMIALS 1. RN LEHNER FRANZ ftept.A path. the of Introduction rmr 5x,4L4;Scnay3B0 53 . 05E35 30B70, Secondary ; 46L54 05Axx, Primary 1 valuation ials. osdrdi h literature, the in considered rew ics Gessel- discuss we erse safnto ntestof set the on function a is cumulants. s e’ hoyo con- of theory let’s tos hc su oa to up is which ctions, fpwr fcontinued of powers of convolution of theory l rbeso various of problems n ro ntrsof terms in proof a e polynomials. l sbe rediscovered been as in aeas been also have tions n ffe probabil- free of ent sso ohaspects both of esis nieysuidin studied ensively rties. hr nbrhand birth on thers lnsi em of terms in ulants fMtknpaths. Motzkin of fcniudfrac- continued of ninterpretation an ’ n Viennot’s and t’s ro orthogonal of ur emnnscan terminants fthe of s Z 2 pair A . paths, z 2 FRANZ LEHNER

1. Starting point and end point lie on the x-axis. 2. The y-coordinates of all points are nonnegative. 3. In each step, the x-coordinate is incremented by one.

Thus a path of length n will start at some point (x0, 0) and end at (x0 + n, 0). The valuations will be independent of the x-coordinates of the points. Therefore the x- coordinates are redundant and we can represent a path π by the sequence of its y- coordinates (π(0), π(1),...,π(n)). We call a path irreducible if it does not touch the x-axis except at the start and at the end. Every path has a unique factorization into irreducible ones.

We will be concerned with two types of paths.

Deﬁnition 2.2. (i) A Motzkin path of length n is a lattice path starting at P0 = (0, 0) and ending at (n, 0), all of whose y coordinates are nonnegative and whose steps are of the three following types. rising step: (1, 1) horizontal step: (1, 0) falling step: (1, −1) Motzkin paths without horizontal steps are called Dyck paths. We will denote the irr set of Motzkin paths by Mn and the subset of irreducible Motzkin paths by Mn . Motzkin paths are counted by the well known Motzkin numbers. (ii) A Lukasiewicz path of length n is a path starting at (0, 0) and ending at (n, 0) whose steps are of the following types. rising step: (1, 1) horizontal step: (1, 0) falling steps: (1, −k), k > 0 We denote the set ofLukasiewicz paths of length n by Ln and the subset of irreducible irr Lukasiewicz paths by Ln .Lukasiewicz paths form a Catalan family, i.e., they are counted by the Catalan numbers.

The nameLukasiewicz path is motivated by the natural bijection to the Lukasiewicz language, the language understood by calculators using reverse polish notation, which was used by Raney in [23] to give a combinatorial proof of the Lagrange inversion formula. Flajolet’s formula expresses the generating function of weighted Motzkin paths as a continued fraction.

Theorem 2.3 ([9]). Let

(2.1) µn = v(π)

πX∈Mn where the sum is over the set of Motzkin paths π =(π(0) ...π(n)) of length n. Here π(j) is the level after the j-th step, and the valuation of a path is the product of the valuations n of its steps v(π)= 1 vi, the latter being Q 1 if the i-th step rises  (2.2) vi = v(π(i − 1), π(i)) = aπ(i−1) if the i-th step is horizontal  λπ(i−1) if the i-th step falls  Then the generating function ∞ n M(z)= µnz Xn=0 CUMULANTS, LATTICE PATHS, AND ORTHOGONAL POLYNOMIALS 3 has the continued fraction expansion 1 (2.3) M(z)= 2 λ1z 1 − α0z − 2 λ2z 1 − α z − 1 . ..

3. Orthogonal Polynomials

A sequence of (formal) orthogonal polynomials is a sequence of monic polynomials Pn(x) of degree deg Pn = n together with a linear functional µ on the space of polynomials with n moments µ(x )= µn such that µ(Pm(x) Pn(x)) = δmnsn for some coeﬃcients sn. We will 0 always assume that s0 = 1, that is µ(x ) = 1. It is then easy to see that such a sequence satisﬁes a three-term recurrence relation

xPn(x)= Pn+1(x)+ anPn(x)+ λnPn−1(x)

with λ0 = 0. The numbers an, λn are the so-called Jacobi parameters. One veriﬁes easily that the polynomials P (x)= Dn(x) satisfy the orthogonality condition, where n ∆n−1

µ0 µ1 ... µn µ1 µ2 µn+1 . . (3.1) Dn(x)= . . . . µ ... µ n−1 2n−1 1 x ... xn

and

µ0 µ1 ... µn µ1 µ2 µn+1 (3.2) ∆n = ...... µ µ ... µ n n+1 2n

is the n-th Hankel determinant. Coming from an operator theoretic background, the most natural way to make the connection to Flajolet’s formula is perhaps via matrices. The Jacobi matrix model for the moment functional is

α0 1 λ1 α1 1  J = , λ2 α2 1  . . .   ......    that is, if we denote the basis of the vector space by e0, e1,... , with inner product hem, eni = δmnsn, then Jen = en+1 + αnen + λnen−1. Then it is easy to see that n Pn(J) e0 = en and therefore J satisﬁes hJ e0, e0i = µn. Expanding the matrix power yields

µn = J0,i1 Ji1,i2 ...Jin−1,0 i1,...,iXn−1≥0

and because of tridiagonality the sum is restricted to indices |ij+1 − ij| ≤ 1. The sum-

mands J0,i1 Ji1,i2 ...Jin−1,0 can be interpreted as the valuations of the Motzkin paths ′ (0, i1, i2,...,in−1, 0) with weights v(y,y )= Jy+1,y′+1, where Ji,i−1 = λi, Ji,i = αi, Ji,i+1 = 1 and this is exactly Flajolet’s formula (2.1). 4 FRANZ LEHNER

4. Cumulants Cumulants linearize convolution of probability measures coming from various notions of independence. Definition 4.1 ([29]). A non-commutative probability space is pair (A,ϕ) of a (complex) unital algebra A and a unital linear functional ϕ. The elements of A are called (non- n commutative) random variables. The collection of moments µn(a) = ϕ(a ) of such a random variable a ∈A will be called its distribution and denoted µa =(µn(a))n. Thus noncommutative probability follows the general “quantum” philosophy of replacing function algebras by noncommutative algebras. We will review several notions of independence below. Convolution is defined as follows. Let a and b be “independent” random variables. Then the convolution of the distributions of a and b is defined to be the distribution of the sum a + b. In all the examples below, the distribution of the sum of “independent” random variables only depends on the individual distributions of the summands and therefore convolution is well defined and the n-th moment µn(a + b) is a polynomial function of the moments of a and b of order less or equal to n. For our purposes it is sufficient to axiomatize cumulants as follows. Definition 4.2. Given a notion of independence on a noncommutative probability space (A,ϕ), a sequence of maps a 7→ kn(a), n = 1, 2,... is called a cumulant sequence if it satisfies

1. additivity: if a and b are “independent” random variables, then kn(a + b)= kn(a)+ kn(b). n 2. homogenity: kn(λa)= λ kn(a). 3. kn(a) is a polynomial in the ﬁrst n moments of a with leading term µn(a). This ensures that conversely the moments can be recovered from the cumulants. We will review here free, classical and boolean cumulants via their matrix models. The reader interested in “q-analogues” is referred to the q-Toeplitz matrix models of [19] and [20] which yield similar valuations onLukasiewicz paths. 4.1. Free Cumulants. Free probability was introduced by Voiculescu in [30] and has seen rapid development since, see [29] and the more recent survey [32].

Deﬁnition 4.3. Given a noncommutative probability space (A,ϕ), the subalgebras Ai ⊆ A are called free independent (or free for short) if

(4.1) ϕ(a1a2 ··· an)=0

whenever aj ∈Aij with ϕ(aj) = 0 and ij =6 ij+1 for j =1,...,n − 1. Elements ai ∈A are called free if the unital subalgebras generated by ai are free. Existence of free cumulants with the properties of definition 4.2 was proved already in [30]. A beautiful systematic theory of free cumulants was found by Speicher in his combinatorial approach to free probability via non-crossing partitions [24]; see [17] for an explanation why noncrossing partitions appear. The explicit computation involves generating functions as follows. ∞ n Theorem 4.4 ([31]). Let M(z) =1+ n=1 µnz be the ordinary moment generating ∞ n function and C(z) =1+ n=1 cnz beP the function implicitly defined by the relation C(zM(z)) = M(z). ThenP the coefficients cn satisfy the requirements of definition 4.2 and are called the free or non-crossing cumulants and C(z) is the cumulant generating function. The moments and the free cumulants are related explicitly by the following combinatorial formula (see [23, 8] for the formulation in terms ofLukasiewicz language, and [24] for CUMULANTS, LATTICE PATHS, AND ORTHOGONAL POLYNOMIALS 5 non-crossing partitions). Define a valuation onLukasiewicz paths by putting the following weights on the steps

v(y,y − k)= ck+1 k ≥ 0 (4.2) v(y,y +1)=1, then

(4.3) µn = v(π).

πX∈Ln Another interpretation of this relation is the Fock space model of Voiculescu [31] (see [13] for a simpler proof). It involves Toeplitz operators as follows. Let S be the forward shift on ℓ2(N0), i.e., Sen = en+1. A (formal) Toeplitz operator is a linear combination ∗ of powers of S and of its adjoint, the backward shift S en = en−1. The linear functional ω(X) = hXe0, e0i is called the vacuum expectation. Let cn be the free cumulants of the moment sequence (µn) as deﬁned above and set ∞ ∗ n (4.4) T = S + cn+1S . Xn=0 n Then ω(T )= µn and writing T in matrix form

c1 1 c2 c1 1  T = c3 c2 c1 1  . . .   . .. ..   we can expand the matrix product and obtain n n ω(T )= hT e0, e0i = T0,i1 Ti1,i2 ...Tin−1,0 i1,...,iXn−1≥0 where Tij =6 0 only for j ≥ i − 1. Again this can be interpreted as a sum over lattice paths which this time turn out to beLukasiewicz paths with weights (4.2) coming from the matrix entries Ti,i+1 =1 Ti,i−k = ck+1, k ≥ 0 and we obtain the sum (4.3). There are multivariate generalizations of this Toeplitz model, see [21]. 4.2. Classical Cumulants. Classical cumulants linearize convolution of measures and can be deﬁned via the Fourier transform.

xz ∞ µn n Deﬁnition 4.5. Let F (z)= µ(e )= n=0 n! z be the exponential moment generating ∞ κn n function. Let K(z) = log F (z)= n=1Pn! z be its formal logarithm. The coeﬃcients κn are the classical cumulants of theP functional µ. There is a model on bosonic Fock space for classical cumulants which is analogous to the Toeplitz model (4.4). Let D, x be annihilation and creation operators which satisfy the canonical commutation relations (CCR): [D, x] = 1, i.e., on the Hilbert space with basis en and inner product hem, eni = n!δm,n, set Den = nen−1, xen = en+1. Then D and x are adjoints of each other. k n Denote again by ω the vacuum expectation ω(T )= hT e0, e0i, i.e. ω(x D )=0 ∀n =06 and ω(f(x)) = f(0). Then the Fourier-Laplace transform of ∞ κ T = D + n+1 xn n! Xn=0

is ∞ zT κn zn ω(e )= e n=1 n! . P 6 FRANZ LEHNER

In the basis {en}, T has matrix representation

κ1 1 κ2 κ1 2  κ3 Tˆ = 2! κ2 κ1 3  κ4 κ3   3! 2! κ2 κ1 4   . . .   . .. ..   e.g. for the stnadard Gaussian distribution we have κ2 = 1 and higher cumulants vanish, so the model is tridiagonal 0 1 10 2  1 0 3  . . .   ......    and this is the well known Jacobi operator matrix for the Hermite polynomials. Similar to the formula above we get the following combinatorial sum for the moments of T :

µn = v(π)

πX∈Ln with valuation κ v(y,y − k)= k+1 k ≥ 0 (4.5) k! v(y,y +1)= y +1

4.3. Boolean Cumulants. Boolean cumulants linearize Boolean convolution which was introduced in [25] (compare also [33], [34]) and is a special case of conditional independence ([5], [4]). In the theory of random walks boolean cumulants arise as ﬁrst return probabilities.

Deﬁnition 4.6. Given a noncommutative probability space (A,ϕ), the subalgebras Ai ⊆ A are called boolean independent if

(4.6) ϕ(a1a2 ··· an)= ϕ(a1) ϕ(a2) ··· ϕ(an)

whenever aj ∈Aij with ij =6 ij+1 for j =1,...,n − 1. With this notion of independence, convolution of measures is well deﬁned and the ∞ n appropriate cumulants can be calculated as follows [25]. Let M(z) = n=0 µnz be the ordinary moment generating function. Then P 1 (4.7) M(z)= 1 − H(z) where ∞ n (4.8) H(z)= hnz , Xn=1

where hn are the boolean cumulants of the distribution with moment sequence (µn). Ex- panding (4.7) in a geometric series we ﬁnd that the moments can be expressed as Cauchy convolution n

µn = hi1 hi2 ··· hir Xr=1 i1+···X+ir=n ij ≥1 CUMULANTS, LATTICE PATHS, AND ORTHOGONAL POLYNOMIALS 7 and from Theorem 2.3 and the fact that every Motzkin path has a unique factorization into irreducible paths, it follows that

(4.9) hn = v(π) irr π∈MXn

where v is the valuation (2.2). This can also be seen by comparing the continued fraction expansion (2.3) with (4.7). In terms of the Jacobi parameters the boolean cumulant generating function has the continued fraction expansion

2 λ1z H(z)= α0z + 2 . λ2z 1 − α1z − 2 λ3z 1 − α z − 2 . ..

Similarly, we can express the boolean cumulants in terms of free cumulants (resp. classical cumulants)

(4.10) hn = v(π) irr πX∈Ln

using the valuations (4.2) (resp. (4.5)).

5. A Formula for Free Cumulants

Theorem 5.1. For n ≥ 2, we have the following formula for the free cumulant cn in terms of the Jacobi parameters:

(−1)|π|0−1 n − 1 (5.1) c = v(π). n n − 1 |π| πX∈Mn 0

Here |π|0 is the number of returns to zero of the path and v is the valuation (2.2) from Flajolet’s formula. Note that the path consisting of n horizontal steps does not contribute.

Proof. The proof consists in comparing the expressions for the moments and free cumulants in terms of the Boolean cumulants (4.8), which themselves represent the sum over irreducible Motzkin paths (4.9). We have then by the multinomial formula

n µn = [z ]M(z) n = [zn] H(z)m mX=0 n n m k1 kn k1+2k2+···+nkn . = [z ] h1 ··· hn z k1,...,kn mX=0 k1+k2+X···+kn=m

k1 + ··· + kn k1 kn = h1 ··· hn k1,...,kn k1+2k2+X···+nkn=n 8 FRANZ LEHNER

The cumulants can be expressed in terms of the moments with the help of Lagrange’s inversion formula [7]: 1 1 c = − [zn] n n − 1 M(z)n−1 1 = − [zn](1 − H(z))n−1 n − 1 1 = − [zn](1 − h z −···− h zn)n−1 n − 1 1 n

1 n (n − 1)! k1 kn k1+2k2+···+nkn = − [z ] (−h1) ··· (−hn) z n − 1 k0! ··· kn! k0+k1+X···+kn=n−1

1 k1+···+kn n − 1 k1 kn = − (−1) h1 ··· hn n − 1 n − 1 − k1 −···− kn,k1,...,kn k1+2k2+X···+nkn=n k1

k1+···+kn−1 (−1) n − 1 k1 + ··· + kn k1 kn = h1 ··· hn n − 1 k1 + ··· + kn k1,...,kn k1+2k2+X···+nkn=n k1

Remark 5.2. Actually by the same argument we can express free cumulants in terms of classical cumulants by simply replacing Motzkin paths byLukasiewicz paths with valuation (4.5). However, there are cancellations in the sum. There are also lots of cancellations when expressing free cumulants in terms of free cumulants themselves usingLukasiewicz paths with valuation (4.2) (only one term survives). Note however that there are no can-

cellations in the sum (5.1), since |π|0 is equal to the number of α0’s and λ1’s and therefore every monomial appears always with the same sign.

6. Hankel determinants

In this section we survey some formulas expressing the Jacobi parameters an and λn in terms of cumulants. It is well known (see e.g. [2]) and can be readily deduced from (3.1), that the Jacobi parameters can be expressed in terms of Hankel determinants and Hankel minors of the moments as follows: ∆n−2∆n λn = 2 ∆n−1 and ∆˜ n ∆˜ n−1 an = − , ∆n ∆n−1 where µ0 µ1 ... µn−1 µn+1 µ1 µ2 µn µn+2 ∆˜ n = ...... µ µ ... µ µ n n+1 2n−1 2n+1

We are therefore interested to express the minors of the inﬁnite Hankel matrix Hµ = [µi+j]i,j≥0 in terms of cumulants. Namely, given ﬁnite sequences of indices i1, i2,...,ip CUMULANTS, LATTICE PATHS, AND ORTHOGONAL POLYNOMIALS 9 and j1, j2,...,jp we will denote

µi1+j1 µi1+j2 ... µi1+jp

i i ... ip µi2 j1 µi2 j2 ... µi2 j (6.1) H 1 2 = + + + p j1 j2 ... jp ...

µip+j1 µip+j2 ... µip+jp

For example,

0 1 ... n 0 1 ... n − 1 n ∆ = H ∆˜ = H . n 0 1 ... n n 0 1 ... n − 1 n +1

The main tool is the so-called Gessel–Viennot theory, see [27, 12, 15, 18]. Let Γ be a weighted graph, that is a graph with vertices V and edges E together with a valuation v : E → R on the edges, where R is some (commutative) ring. The valuation of a path ω = (s0,...,sm) is the product of the valuations of the steps v(ω) = v(s0,s1) v(s1,s2) ··· v(sm−1,sm). Select two sets of distinct vertices Ai, Bi ∈ V , i =1,...,n and suppose that the set of paths

Ωij = {ω =(s0,...,sm): s0 = Ai,sm = Bj, v(ω) =06 } is ﬁnite for every pair of indices (i, j). We will say that two paths intersect if they have a vertex in common. We will not care about crossings of paths, that is, crossings of edges in a graphical representation. Deﬁne a matrix

aij = v(ω), 1 ≤ i, j ≤ n.

ωX∈Ωij Then the determinant of this matrix has the following combinatorial interpretation.

Proposition 6.1 ([27] Prop. IV.2).

(6.2) det[aij]n×n = sign(σ) v(ω1) ··· v(ωn)

(σ;ωX1,...,ωn) S where the sum is over all permutations σ ∈ n, ωi ∈ Ωi,σi and all non-intersecting (but possibly crossing) paths ωi ∈ Ωi,σ(i). We have seen various examples above where the n-th moment is equal to the sum of the valuations of paths of length n. All our valuations are translation independent and we will interpret the (i, j) entry of Hµ as a sum µi+j = v(ω) over the set of paths starting at (−i, 0) and ending at (j, 0). The minor (6.1) is thereforeP equal to

i1 i2 ... ip sign(σ) H = (−1) v(ω1) v(ω2) ··· v(ωp) j1 j2 ... jp σX∈Sp ω1,ωX2,...,ωp

where the sum extends over all permutations σ of the indices and all nonintersecting configurations of paths ωk starting at (−ik, 0) and ending at (jσ(k), 0). Because the vertices are ordered, the sign is (−1)sign(π) = (−1)K, where K is the number of crossings of the configuration corresponding to π. For example, for the full Hankel determinant from (3.2), where the moments are interpreted as sum over Motzkin paths as in Theorem 2.3, there is only one non-intersecting configuration whose picture is (for n = 4) 10 FRANZ LEHNER

0, 1, 2, 3, 4 H 0, 1, 2, 3, 4 n n−1 2 and therefore ∆n = λ1 λ2 ··· λn−1λn. Note that in this example the point (0, 0) is considered as a path of length zero and is not allowed to lie on any of the other paths. In contrast, in the case ofLukasiewicz paths with valuations (4.2) and (4.5), there may occur crossings of paths and there are many contributing terms with diﬀerent signs. It is also possible that cancellations occur in the sum, e.g.

  2 2   v = −c1c2c3 = −v             0 1 2 are cancelling contributions to H . Therefore, and because the sums are com- 0 1 3 plicated, the formulae are of rather limited value. References [1] Accardi, L. and Bo˙zejko, M., Interacting Fock spaces and Gaussianization of probability measures, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), 663–670. [2] Akhiezer, N. I., The classical moment problem and some related questions in analysis, Hafner Pub- lishing Co., New York, 1965. [3] Akiyama, M. and Yoshida, H., The orthogonal polynomials for a linear sum of a free family of projections, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 (1999), 627–643. [4] Bo˙zejko, M., Leinert, M. and Speicher, R., Convolution and limit theorems for conditionally free random variables, Pacific J. Math. 175 (1996), 357–388. [5] Bo˙zejko, M. and Speicher, R., ψ-independent and symmetrized white noises, Quantum probability & related topics, World Sci. Publishing, River Edge, NJ, 1991, pp. 219–236. [6] Cohen, J. M. and Trenholme, A. R., Orthogonal polynomials with a constant recursion formula and an application to harmonic analysis, J. Funct. Anal. 59 (1984), 175–184. [7] Comtet, L., Advanced combinatorics, D. Reidel Publishing Co., Dordrecht, 1974. [8] Cori, R., Words and trees, Combinatorics on Words (Lothaire, M., ed.), Encyclopedia of mathematics, vol. 17, Addison-Wesley Publishing Co., Reading, Mass., 1983, pp. 213–227. [9] Flajolet, P., Combinatorial aspects of continued fractions, Discrete Math. 32 (1980), 125–161. [10] Flajolet, P. and Guillemin, F., The formal theory of birth-and-death processes, lattice path combinatorics and continued fractions, Adv. in Appl. Probab. 32 (2000), 750–778. [11] Gerl, P., Continued fraction methods for random walks on N and on trees, Probability measures on groups, VII (Oberwolfach, 1983), Lecture Notes in Mathematics, vol. 1064, Springer, Berlin, 1984, pp. 131–146. [12] Gessel, I. M. and Viennot, X. G., Determinants, paths, and plane partitions, Preprint, 1989. [13] Haagerup, U., On Voiculescu’s R- and S-transforms for free non-commuting random variables, Free probability theory (Waterloo, ON, 1995), Amer. Math. Soc., Providence, RI, 1997, pp. 127–148. [14] Ismail, M. E. H., Masson, D. R., Letessier, J. and Valent, G., Birth and death processes and orthogonal polynomials, Orthogonal polynomials (Columbus, OH, 1989), Kluwer Acad. Publ., Dordrecht, 1990, pp. 229–255. CUMULANTS, LATTICE PATHS, AND ORTHOGONAL POLYNOMIALS 11

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Franz Lehner, Institut fur¨ Mathematik C, Technische Universitat¨ Graz, Steyrergasse 30, A-8010 Graz, Austria E-mail address: [email protected]