A Study of Viennot's Combinatorial Models of Orthogonal Polynomials

Home , Orthogonal polynomials, Theodore Motzkin

A Study of Viennot’s Combinatorial Models of Orthogonal Polynomials

Krista L. Smith with Advisor Dr. Gabor Hetyei

Department of Mathematics and Statistics The University of North Carolina at Charlotte 9201 University City Blvd. Charlotte, NC 28223

April 25, 2014

1 Introduction

This research project explores the subject of Orthogonal Polynomial Se- quences and algebraic methods for determining their corresponding moments using Xavier Viennot’s numerous combinatorial models. Orthogonal polynomials include Hermite, Laguerre, and Jacobi among others. Throughout this project, we will keep the focus on the Chebyshev and Laguerre Polynomial Sequences. Orthogonal Polynomial Sequences (OPS) are bases of the vector space of polynomials that are orthogonal with respect to an inner product, deﬁned in terms of a linear functional, called the moment functional. For Chebyshev polynomials of the 1st kind, the moment functional is given by

Z 1 1 L [xn] = xn √ dx. (1) 2 −1 1 − x The inner product of two polynomials can be found by substituting the product of the two polynomials into the moment functional. The moment functional is uniquely determined by the moments L [xn] obtained by substituting the powers of x into the moment functional. An OPS must satisfy the following conditions for n ≥ 1:

P−1(x) = 0

1 P0(x) = 1

L [PmPn] = 0, m 6= n Favard’s Theorem, a key result of the theory of OPS, characterizes their monic variants as satisfying a three-term recurrence:

Pn+1 = (x − bn)Pn − λnPn−1. (2) Favard’s Theorem states that if the above conditions are met, then the moment functional L is quasi-deﬁnite and Pn(x) is a monic OPS. This relation holds for any orthogonal polynomial sequence taken with respect to a positive weight function.

2 Preliminaries

2.1 Xavier Viennot Emeritus Research Director at the French government’s National Center for Scientific Research, Xavier Viennot’s work focuses on combinatorics with applications to pure and applied mathematics, computer science and physics. His research mainly exists as scans of French language lecture notes posted on the author’s website, although a brief overview of these notes was pub- lished in English in 2003 by Dennis Stanton in his book, Orthogonal Poly- nomials and Special Functions. We are mainly interested in the two chapters of the lecture notes entitled Moments et rećurrence lineáire and Moments de familles particulie`res de polynoˆmes orthogonaux, translating as Moments and linear recurrence and Particular moments of families of orthogonal polynomials, re- spectively.

2.2 Quadratic Forms Given a symmetric bilinear form F on a real vector space V , we deﬁne a map Q : V −→ R by Q(v) = F (v, v); where Q is the quadratic form associated with the symmetric bilinear form F [1]. Quadratic forms are homogeneous quadratic polynomials in n variables. The term quadratic form references Q being a homogeneous quadratic function of the coordinates (where each term has the same degree of two) [1]. Quadratic forms have matrix representations which are symmetric and unique. A quadratic form that is positive (nonnegative) on all vectors is positive (semi) deﬁnite. Every real, symmetric matrix represents a quadratic form,

2 and vice versa. Notice that the quadratic function, ax2 + bx + c is not a quadratic form in a one variable case because it is not homogeneous. In general, the positive (semi) definite property of a quadratic form may be characterized in terms of the leading principal minors. An analogous description exists for moment functionals. A moment functional L is positive definite if L [π(x)] > 0 for every polynomial π(x) that is not identically zero and is nonnegative for all real x [3]. Let L be positive definite. Then L has real moments and a corresponding OPS consisting of real polynomials exists.

2.3 Polynomials as a Vector Space

The set of polynomials with coefficients in F is a vector space over F. Vector addition and scalar multiplication are defined in the obvious manner. If the degree of the polynomials is restricted to polynomials with degree less than or equal to n, then we have a vector space with dimension n + 1. The concept of an orthogonal basis is applicable to a vector space V (over any field) equipped with a symmetric bilinear form. In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. Every vector space equipped with a symmetric bilinear form has an orthogonal basis which may be found using the Gram-Schmidt Pro- cess. Orthogonal Polynomial Sequences are orthogonal bases with respect to a special symmetric bilinear form, known as the moment functional (see section 1).

2.4 Orthogonal Polynomials

Deﬁnition 2.1. A sequence Pn(x) from n = 0 to ∞ is called an Orthogonal Polynomial Sequence with respect to a moment functional L provided for all nonnegative integers m and n:

(i) Pn(x) is a polynomial of degree n (ii) L [Pm(x)Pn(x)] = 0 for m 6= n 2 (iii) L [Pn (x)] 6= 0. Orthogonal polynomials include Hermite, Laguerre, and Jacobi among others. Orthogonal Polynomial Sequences (OPS) are bases of the vector space of polynomials that are orthogonal with respect to an inner product, deﬁned in terms of a linear functional, called the moment functional. All orthogonal polynomials can be expressed in terms of their moments, and their monic variants satisfy Favard’s Theorem [3]:

3 Pn(x) = (x − cn)Pn−1(x) − λnPn−2(x). (3) Conversely, if a sequence of polynomials satisﬁes a recurrence in the form of equation (3) then it is an Orthogonal Polynomial Sequence. Moments are computed recursively.

2.4.1 Chebychev Polynomials of the 1st Kind Chebyshev polynomials appear in many diﬀerent branches of mathematics beyond combinatorics, including diﬀerential equations, geometry, statistics, numerical analysis, number theory and approximation theory, as well as in physics. Of the 13 classical OPS listed by Abramowitz and Stegun [4], 6 are distinct Chebyshev polynomial sequences. The Chebyshev polynomials of the 1st kind are derived by making a change of variables as given by:

Tn(x) = cos(nθ), (4) where: x = cos(θ). Favard’s recurrence for the monic variant of the Chebyshev polynomials Pn(x) with unique constants bn and λn is deﬁned as:

for n = 1 1 P (x) = xP (x) − P (x). (5) n+1 n 2 n−1 for n ≥ 2 1 P (x) = xP (x) − P (x). (6) n+1 n 4 n−1

We see that all bn values are zero in Favard’s recurrence for the Cheby- chev polynomials. We also see that there are two possible values for λ. We will see these constants again when we explore Viennot’s combinatorial model for calculating OPS moments, where we have used the Chebyshev sequence as an example. The generating series for Chebyshev of the 1st kind:

X 1 − xt T (x) = T tn = . (7) n n 1 − 2xt + t2 n≥0 The ﬁrst 6 terms in the normalized Chebyshev polynomial sequence are:

4 Figure 1: First 6 terms of the normalized Chebyshev polynomial sequence of the 1st kind.

P0(x) = 1

P1(x) = x 1 P (x) = x2 − 2 2 3 P (x) = x3 − x 3 4 1 P (x) = x4 − x2 + 4 8 5 5 P (x) = x5 − x3 + x 5 4 16

Classically, the moments of the Chebyshev polynomials sequence of the 1st kind can be determined using the moment functional from equation (1):

Z 1 1 L [xn] = xn √ dx. 2 −1 1 − x We determine the ﬁrst six moments as the following:

Z 1 0 1 µ0 = L [x ] = √ dx = π 2 −1 1 − x Z 1 1 x µ1 = L [x ] = √ dx = 0 2 −1 1 − x

5 Z 1 2 2 x π µ2 = L [x ] = √ dx = 2 −1 1 − x 2 Z 1 3 3 x µ3 = L [x ] = √ dx = 0 2 −1 1 − x Z 1 4 4 x 3π µ4 = L [x ] = √ dx = 2 −1 1 − x 8 Z 1 5 5 x µ5 = L [x ] = √ dx = 0 2 −1 1 − x We notice that all odd moments are zero for the Chebyshev OPS, as they should be for an odd function integrated over an interval that is symmetric to the origin.

3 Viennot’s Combinatorial Models for Moments and Recurrence

Viennot’s chapter entitled Moments and linear recurrence introduces an alternative method to calculating the moments of orthogonal polynomial sequences. The traditional approach is to evaluate the inner product, given by condition ii of Deﬁnition 2.1, which is manageable for the ﬁrst few terms in any OPS. But as we approach higher order polynomials in the sequence, the terms contained in each polynomial grows considerably. Calculating the inner product of a ninth and tenth term polynomial (including cross terms as this is a product) becomes quite a formidable task. Viennot’s model is based on the constant terms bn and λn within Favard’s Theorem discussed in the Introduction, that are unique to each monic OPS.

3.1 Viennot’s Favard Paths Viennot has devised a combinatorial model for determining the next polynomial in the OPS using weighted lattice paths, as an alternative to using the recurrence relation presented by Favard’s Theorem. Although the recurrence relation for the Chebyshev polynomials is very straightforward, this is not always the case. For instance, the recurrence relation for determining the next polynomial in the sequence can be very cumbersome when working with the Legendre polynomial series. For this reason, Viennot has presented

6 Figure 2: The weighted steps for Favard Path of Chebyshev polynomials of the 1st kind. a combinatorial model as a simpler alternative to troublesome recurrence calculations. In Favard paths, three steps are allowed: the North East step (1, 1) of weight x, the North step (1, 0) of weight −bn and the North North step (2, 0) of weight −λk. Since we are working with Chebyshev polynomials in this case, the constant −bn = 0, therefore the Favard paths will not include any North steps. We can use Viennot’s Favard paths to reconstruct the recurrence relation of equation (2):

Pn+1 = (x − bn)Pn − λnPn−1 −→ Pn+1 = xPn − λnPn−1. (8)

The ﬁrst 5 polynomials are:

P0 = 1

P1 = x 1 P = xP − λ P = x2 − (9) 2 1 n 0 2 3 P = xP − λ P = x3 − x (10) 3 2 n 1 4 1 P = xP − λ P = x4 − x2 + (11) 4 3 n 2 8

The corresponding Favard paths are shown in Figure 3.

7 Figure 3: The weighted steps for Favard Paths of Chebyshev Polynomials of the 1st kind.

3.2 Motzkin Numbers Theodore Motzkin was an Israeli-American mathematician and Professor of Mathematics at UCLA in the 1960’s who made numerous discoveries during his career, including the sequence known as the Motzkin numbers Mn. Among other applications, the Motzkin numbers are a famous result about the total number of ways to draw non-intersecting chords between n points on a circle. The generating function for the Motzkin sequence is given by: √ X 1 + t − 1 − 2t + 3t2 M tn = , (12) n 2t t≥0

with recurrence relation:

3(n − 1)M + (2n + 1)M M = n−2 n−1 . (13) n n + 2

8 Motzkin numbers are closely related to the Catalan numbers (see next section): for n ≥ 0 n X2 n M = C . (14) n 2k k k=0

The ﬁrst eight terms in the Motzkin sequence are: 1, 1, 2, 4, 9, 21, 51, 127.

3.3 Catalan Numbers Catalan numbers, introduced around 1850 by Belgian and French mathematician Eug`ene Charles Catalan, arise frequently in combinatorics, as evidenced by over 60 diﬀerent interpretations in Richard P. Stanley’s second volume of Enumerative Combinatorics. Additionally, Stanley’s Catalan Addendum [7] contains over 200 problems involving Catalan numbers.

1 2n C = , (15) n n + 1 n where the generating function is deﬁned by:

∞ X n c(x) = Cnx . (16) n=0 The ﬁrst 6 terms in the natural number sequence are: 1, 1, 2, 5, 14, 42. The Catalan numbers are well-known to represent the total number of Dyck paths whereas the Motzkin numbers represent the total number of Motzkin paths (section 3.4).

3.4 Viennot’s Motzkin Paths Motzkin paths are lattice paths from the point of origin (0, 0) to the point (n, 0) in which the path lies strictly above, but may touch, the x-axis. The allowed steps are the rise step (NE): diagonal (1, 1), the fall step (SE): diagonal (−1, 1) and the level step (E): (1, 0). A Dyck path is a lattice path using the rise step: diagonal (1, 1) and the fall step: diagonal (1, −1) from the point of origin (0, 0) to the point (2n, 0) in which the path lies strictly above, but may touch, the horizontal axis. Since there are no horizontal steps, there must be n number of NE steps and n number of SE steps, and therefore, the x coordinate of any point on the x axis must be even.

9 The number of Dyck paths of length 2n can be expressed as a Cata- lan number, according to equation (15). Dyck paths are a special type of Motzkin path, where both the rise and fall steps are allowed, but the horizontal step is not. In the special case of an OPS with bk = 0 in the Fundamental Recurrence relation, the OPS can be expressed as a weighted Dyck path. The combinatorial model presented by Viennot in this section therefore does not apply to all OPS. To compute the moments, the classical methods are to evaluate the integral formula using integration by parts (if given), or the method described in the proof of Favard’s Theorem. Either method involves recursive calculations, and closed form formulas are shown by proving a conjecture by induction. As an alternative, Viennot presents his weighted Motzkin path model, where understanding the underlying combinatorics often allows for proving closed form formulas directly. Similarly, whether an OPS is found using the Gram-Schmidt process or Favard’s recurrence, we obtain the polynomials in a recursive fashion. To give an overview of this process at a glance, see section /refsec:Favard. Motzkin paths are used in a weighted path model created by Viennot as an alternative method for determining the moments of Orthogonal Polyno- mial Sequences using moment functionals, which involve integration. Vien- not’s model is based on fractions and sums, with each allowed step being assigned a weight corresponding to its known values in the recurrence relation of equation (2). We choose the Chebyshev polynomials to illustrate this method.

Figure 4: The weighted steps for Motzkin Paths of Chebyshev polynomials of the 1st kind, where w signiﬁes the weight.

For weights of the Chebyshev polynomials of the 1st kind, see Figure 4. We can express the moment µn the total weight of all weighted Motzkin

10 Figure 5: All possible Motzkin paths from (0, 0) to (4, 0) for µ4. When all bn = 0, we only consider Dyck paths, as described in section 3.4. paths from (0, 0) to (n, 0). For example, if we would like to ﬁnd the moment of µ4, as shown in Figure 5, we will look to ﬁnd all possible Motzkin paths from (0, 0) to (4, 0), where the total weight of each path is the product of each step. There are 9 possible Motzkin paths from (0, 0) to (4, 0), and recalling the sequence of Motzkin numbers from the previous section, 9 is the Motzkin number M4. The sum of the weights of the 9 paths is the following:

4 2 2 2 2 2 b0 + λ1 + b1λ1 + b0λ1 + b0λ1 + λ1b0 + b0λ1b0 + b0b1λ1 + λ2λ1

4 4 2 2 =⇒ b0 + 3b0λ1 + 2b0b1λ1 + b1λ1 + λ1 + λ1λ2. For the Chebyshev polynomials of the 1st kind, 1 1 λ = − , λ = − , b = 0. 1 4 2 4 k

Therefore the total weight of all Motzkin paths for µ4 is:

1 1 1 1 3 − − + − − = . (17) 2 4 2 2 8

11 Figure 6: All possible Motzkin paths from (0, 0) to (3, 0) for µ3 of the Chebyshev polynomials of the 1st kind.

Note: Viennot’s moments are normalized such that µ0 = 1. If we use another moment functional that differs by a constant factor, then all moments need to be multiplied by µ0. Here, that constant factor is π. We can immediately see that when bk = 0, the only Motzkin paths contributing to the µ4 are paths B and I, which are the only paths that do not contain the horizontal step. This leads to Viennot’s next discussion of Chebyshev polynomials and Dyck paths. The calculations for the previous moments are included in Viennot’s work. However, we did find an honest mistake in the calculation of µ3 as outlined by Viennot. Viennot’s formula as given on page I−14 of Chapitre I−Moments et rećurrence lineáire: 3 µ3 = b0 + 3b0λ1. (18)

The third moment corresponds to the fourth Motzkin number, M3, so there are a total of 4 Motzkin paths, as shown in Figure 6. However, adding up the weights of these 4 paths actually gives the following:

3 µ3 = b0 + 2b0λ1 + b1λ1, (19) which is of diﬀerent form than what Viennot has written.

4 Chebyshev Polynomials of the 1st and 2nd Kind and Dyck Paths

Here we take a closer look at calculating the moments of the Chebyshev polynomials of the 1st and 2nd kind using weighted lattice paths. Since

12 both OPS have a recurrence relation with bn = 0 (see section 3.4), we will be working with Dyck paths.

4.1 Chebyshev Polynomials of the 2nd Kind and Dyck Paths The Chebyshev polynomials of the 2nd kind are also polynomials in x of √degree n, but orthogonal with respect to a diﬀerent weight function (w = 1 − x2 ). They are derived by making a change of variables as given by:

sin((n + 1)θ) U (x) = , (20) n sin θ where: x = cos(θ). The generating series for Chebyshev polynomials of the 2nd kind:

X n 2 −1 Un(x) = Un(x)t = (1 − 2xt + t ) , (21) n≥0 and the recurrence relation:

Un+1 = 2xUn(x) − Un−1(x), (22) with monic polynomials Nn(x) given by: 1 N (x) = U (x). (23) n 2n n The ﬁrst 6 terms in the monic Chebyshev Orthogonal Polynomial sequence of the 2nd kind are:

N0(x) = 1

N1(x) = x 1 N (x) = x2 − 2 4 1 N (x) = x3 − x 3 2 3 1 N (x) = x4 − x2 + 4 4 16 3 N (x) = x5 − x3 + x 5 16

13 Figure 7: The ﬁrst 6 terms of the normalized Chebyshev polynomials of the 2nd kind.

The ﬁrst 6 moments can be found using the moment functional described in section 1, and substituting the weight function for Chebyshev polynomials of the 2nd kind:

Z 1 n np 2 µn = L [x ] = x 1 − x dx. (24) −1 We see that all odd moments are zero, as expected:

Z 1 0 p 2 π µ0 = L [x ] = 1 − x dx = −1 2 Z 1 1 p 2 µ1 = L [x ] = x 1 − x dx = 0 −1 Z 1 2 2p 2 π µ2 = L [x ] = x 1 − x dx = −1 8 Z 1 3 3p 2 µ3 = L [x ] = x 1 − x dx = 0 −1 Z 1 4 4p 2 π µ4 = L [x ] = x 1 − x dx = −1 16 Z 1 5 5p 2 µ5 = L [x ] = x 1 − x dx = 0 −1

14 As previously stated, the moments of Chebyshev polynomials can be determined using weighted Dyck paths. There are two reasons for this, the ﬁrst being that the integral of any odd function (for example, the sine function) over an interval symmetric to the origin is always zero:

Z 1 µ2n+1 = sin(x) dx = 0. (25) −1

The second reason is that when bn = 0 (see section 3.4), we only use Dyck paths, which cannot end at (2n + 1, 0). This implies:

µ2n+1 = 0. (26) In the case of the Chebyshev polynomials of the 2nd kind, their moments can be determined directly from the Catalan numbers, Cn. Viennot’s model gives the following formula:

1 1 2n 1 µ = = C (27) 2n 4n n + 1 n 4n n

and µ2n+1 = 0 for all n ≥ 0.

Equation (26) counts the number of Dyck paths and factors out the down steps, λk, for which Chebyshev polynomials of the 2nd kind are all equal to 1/4. Therefore, all Dyck paths of equal length have equal weight (see Figure 8), since every NE step has the same weight (1) and every SE step has the same weight (1/4).

Figure 8: For Chebyshev polynomials of the 2nd kind, all Dyck paths of equal length have the same weight.

Note: Not all down steps for the 1st kind share the same value (see Figure 9), which is why it is generally easier to construct combinatorial models for

15 Chebyshev polynomials of the 2nd kind (see section 4.2). The moments obtained from equation (26) diﬀer by a factor of π/2 from the moments in the classical literature. For example, the classical formula for the fourth moment is:

Z 1 4 4p 2 π µ4 = L [x ] = x 1 − x dx = . (28) −1 16 Substituting n = 2 into Equation (23) gives:

C 2 1 µ = 2 = = . (29) 2n 42 16 8 The classical moment functional for the ﬁrst moment is:

Z 1 0 p 2 π µ0 = L [x ] = 1 − x dx = . (30) −1 2 If we multiply the moments obtained through Viennot’s model by the classical moment functional for µ0, we then arrive at the equivalent fourth moment as determined by the classical method:

C 1 π π µ = 2 (µ ) = = . (31) 4 42 0 8 2 16 Comparing equation (25) to equation (28), we see how easily one can obtain the moments of the Chebyshev polynomials of the 2nd kind once we realize their relationship to Dyck paths. We do not need to ﬁnd all Dyck paths for any of the moments, since the Catalan number sequence already tells us how many there are. Additionally, only one integral must be calculated, which is the most simple one: µ0.

4.2 Chebyshev Polynomials of the 1st Kind and Dyck Paths Classically, the moments of Chebyshev polynomials of the 1st kind are determined by equation (1). Let us look at an example, µ6:

Z 1 6 6 1 5π µ6 = L [x ] = x √ dx = . (32) 2 −1 1 − x 16 For even moments of the Chebyshev polynomials of the 1st kind, Viennot ﬁnds the following formula:

1 2n µ = . (33) 2n 4n n

16 Figure 9: Two Dyck paths of equal length do not carry the same weight for Chebyshev polynomials of the 1st kind.

This relation does not contain the Catalan number sequence, as does Viennot’s formula for the moments of the 2nd kind (see equation (27)). This is because we are now counting Dyck paths with diﬀerent weights, since the Chebyshev polynomials of the 1st kind have two values for λk, as shown in Figure 9. Chebyshev polynomials of the 2nd kind have only one value for λk (see section 4.1). The weight v(ω) for the ﬁrst Dyck path in Figure 9 is the product of the weight of each step:

1 1 1 v(ω ) = (1)(1) = . (34) 1 4 2 8 But the second Dyck Path of equal length has weight:

1 1 1 v(ω ) = (1) (1) = . (35) 2 2 2 4 For Chebyshev of the 1st kind, all NE steps also have weight 1, but all SE steps have weight 1/4 except for λ1 = 1/2, which is the SE step that touches the axis. From here, we could draw all possible Dyck paths, calculate the weight for each individual path, then sum over all weights to ﬁnd each moment. For this method, we would use just one lattice path (the Dyck path), and allow the weights of each SE step to vary. For µ6, n = 3, and therefore the number of Dyck paths corresponds to the third Catalan number, C3 = 5. These 5 Dyck paths are illustrated in Figure 10. The total weight for µ6 is then given by:

17 Figure 10: The 5 Dyck paths corresponding to µ6. All NE steps have weight 1 and all SE steps have weight 1/4 except for the SE step that touches the axis (1/2).

X 1 1 1 1 1 5 V (ω) = v(ω) = + + + + = . (36) 8 16 16 32 32 16 |ω|=n The Chebyshev moments of the 1st kind are normalized in the same fashion as the 2nd kind (see section 4.1), and one must simply multiply by µ0 to recover the exact value initially obtained from computation of the moment functional in equation (32).

Z 1 0 1 µ0 = L [x ] = √ dx = π. (37) 2 −1 1 − x Therefore, 5π µ = µ (V (ω)) = . (38) 6 0 16 However, Viennot found an alternative to this approach: we just use the same weight (1/4) for each SE step, and allow the lattice paths to vary instead, as described below. Another form of equation (33) is the following:

18 Figure 11: The total number of lattice paths from (0, 0) to (6, 0) consisting of ﬂipping two consecutive.

n 1 X µ = 2k(ω), (39) 2n 4 ω where k(ω) is the number of returns of the x axis, and the summation is over all Dyck paths ω of length 2n. First we assign weight 1/4 to all SE steps then correct the weighting for each SE step touching the x axis. We may then reinterpret each factor of 2 as a possibility to reflect a part of the Dyck path between two consecutive returns to the x axis. These choices may be performed independently. Note that we can take any part of a Dyck path between two of its points on the x axis and reflect just this part about the x axis. This way we may associate 2k variants to the same Dyck path. These variants are all lattice paths from (0, 0) to (2n, 0) however, they are allowed to go below the x axis. Conversely, if we have a lattice path from (0, 0) to (2n, 0) returning to the x axis exactly k times, we may flip up its parts below the x axis, obtaining a unique Dyck path, as shown in Figure 11. Therefore, instead of counting each Dyck path with weight 2k, where k is the number of returns to the x axis, we count all lattice paths from (0, 0) to (2n, 0), using NE and SE steps 2n only. The total number of such lattice paths is n . Hence, for even moments of the Chebyshev polynomials of the 1st kind, Viennot finds the following formula:

19 Figure 12: The 8 lattice paths from Figure 10 can be arranged in the form of a cube of unit length. The Dyck path is located at (1, 1, 1).

1 2n µ = . (40) 2n 4n n Substituting in n = 3, we have:

1 6 5π µ = (µ ) = , (41) 6 43 3 0 16 which is equivalant to the value given by the moment functional in equation (32). As an interesting side note, the 8 lattice paths in red shown in Figure 11 can be arranged into a pattern depicted in Figure 12. There are 8 lattice paths, which is also the number of vertices on a cube. Two orientations are allowed for the paths, to which we can assign the number 0 or 1. Figure 12 shows each path and its corresponding Cartesian coordinates in 3D space. When all 8 coordinates are brought together, they form the k(ω) unit cube. This is an example of the group action Zn , where k = k(ω). In summary, we discussed finding the moments for the Chebyshev Poly- nomial Sequence in this section from three different approaches. The first approach is the classical method using the moment functional. The second approach is summing the different weights of all Dyck paths, and the third approach is to sum equally weighted but different lattice paths.

20 5 Laguerre Polynomial Moments and Permutations

Lastly we will look at a family of Orthogonal Polynomial Sequences, the La- guerre polynomials. Here Viennot shows the relationship between moments of the Laguerre polynomials, increasing binary trees, colored lattice paths and the symmetric permutation group Sn.

5.1 Laguerre Polynomials α The Generalized (or associated) Laguerre polynomials Ln(x) were discovered by Edmond Laguerre (1834 − 1886), and are solutions to the homogeneous diﬀerential equation known as Laguerre’s equation:

xy00 + (α + 1 − x)y0 + λy = 0. (42) In the case when α = 0, the solutions are known as the classic Laguerre polynomials. In all other cases, the solutions are known as the generalized Laguerre polynomials. The generalized Laguerre polynomials are deﬁned as:

dα Lα(x) = (−1)α L (x), (43) n dxα n+α

and form an Orthogonal Polynomial Sequence over the interval (0, ∞) with respect to the weight function xke−x, with inner product deﬁned as:

Z ∞ −x α α α (n + α)! e x Ln(x)Lm(x) dx = , δmn. (44) 0 n!

The generalized Laguerre polynomials satisfy the recurrence relation Ln+α(x) for all n ≥ 1:

α α α (n + 1)Ln+1(x) = (2n + k + 1)Ln(x) − (n + k)Ln−1(x), (45)

α with closed form expression Ln(x):

∞ X 1 dn Lα(x) = (x)−αex xn+αe−x, (46) n n! dxn n=0

and generating function Φ(x, α, k) deﬁned as:

21 Figure 13: First 6 terms of the normalized Laguerre polynomial for the sequence α = 0.

∞ X α n Φ(x, α, k) = Ln(x)k . (47) n=0

For α = 0, the ﬁrst 6 classic Laguerre polynomials are:

0 L0(x) = 1 0 L1(x) = −x + 1 1 L0(x) = (x2 − 4x + 2) 2 2 1 L0(x) = (−x3 + 9x2 − 18x + 6) 3 6 1 L0(x) = (x4 − 16x3 + 72x2 − 96x + 24) 4 24 1 L0(x) = (−x5 + 25x4 − 200x3 + 600x2 − 600x + 120) 5 120

Classically, the leading coeﬃcients of Ln(x) are: (−1)n . (48) n!

However, Viennot works with the monic polynomials, `n(x), of the form:

22 Figure 14: The binary tree representing permutation 362154789.

n `n(x) = (−1) n! Ln(x). (49) For Favard’s recurrence (Equation (2)):

bk = 2k + α + 1λk = k(k + α).

5.2 Binary Trees This combinatorial model Viennot constructed for the Laguerre polynomials is based on a bijection that exists between increasing binary trees and the permutations associated with the moments of these polynomials. We present the model for α = 1, but it is possible to generalize to other α. The relation to colored Motzkin paths will be discussed in section /refsec:Colored. Beginning with a permutation, we can construct a binary tree. The binary trees used in Viennot’s combinatorial model are rooted planar binary trees. Non-leaf vertices have at most two children. If a node has a single child, it could be left or right. For example, the permutation π(b) = 362154789 would have the increasing binary tree shown in Figure 13. We see that 1 is centered as the root, and all numbers in the permutation to the left of the root are mapped within the left subtree, and all numbers to the right of the the root are mapped within the right subtree. The binary tree then represents the permutation in increasing numeric order from top to bottom, with the left subtree signifying the numbers in the permutation that come before the parent, and the right subtree signifying the numbers in the permutation following the parent. Therefore, the increasing binary tree is read top to bottom as well as left to right.

23 5.3 Colored Lattice Paths Viennot’s colored lattice paths can classify permutations from their corresponding binary trees. The colored lattice paths have two codes associated with them, the ﬁrst to express the number of choices present at each vertex, and the second, to express the choices that were taken.

Figure 15: The colored Motzkin path representing permutation 362154789, with total choices determined by vertex levels, and selections determined by binary tree placement.

Viennot assigns a red colored horizontal step (E, red) for every parent with only a left child, and assigned a blue horizontal colored step (E, blue) for every parent with only a right child. If the parent has no child, the lattice path has step southeast (SE), and if the parent has two children, the lattice path has step northeast (NE). These colored lattice paths are therefore Motzkin paths, since the level step is allowed and the paths lie strictly above the x-axis. The lattice path associated with permutation 362154789 is shown in Figure 14. Viennot suggests that the permutation representing all possible choices of this lattice path can be determined by simply counting k +1 levels (where the horizontal axis is level 0 and we omit the last vertex) such that a lattice path with n = 9 vertices could then be described by a permutation with n − 1 = 8 numbers. These k + 1 levels that represent the possible choices at each vertex in the lattice path are determined from the construction of the binary tree (see Figure 15), where we begin counting on the point, but stop before we reach the “Extra Step”.

24 Figure 16: The construction of the binary tree representing permutation 362154789, where the number of branches in each step corresponds to the number of choices at each vertex (level k + 1).

The colored Motzkin Paths do not determine the permutation uniquely unless the choices taken at each vertex are specified. Thus, we add a third line that specifies which choices were taken at each vertex. We look to the binary tree of Figure 16, which shows the placement of the number on each branch. To find the choices selected, we begin counting at the root of the tree, and ending after we count the “Extra Step”. Starting from the root and working in increasing numeric order, we count the placement of the child (the leaf) with the parent (the branch) from left to right. If there are three branches, and we place the leaf at the second, then the code we record is 2. If we place it at the first branch, we record 1 and if the third position, we record 3. Therefore, Figure 15 depicts the number of choices possible at each step (represented by the number of branches on the tree) while Figure 16 depicts the choices selected (represented by the placement of the number on the tree). Observe that in this model, from level k ≥ 0 there are ak = k + 1 choices below a NE step and bk = 2(k+1) below a red or blue horizontal step. From level k ≥ 1 there are ck = k + 1 choices below a SE step. Note that the numbers bk are the coefficients in Favard’s recurrence and we have λk = ak−1bk. Since a Motzkin path begins and ends at level 0, we can match each NE step with the next SE step that returns to the same level. The weight of

25 Figure 17: The construction of the binary tree representing permutation 362154789, where the placement of the number on the branch corresponds to the choice selected at each vertex.

this path can be expressed as NE·SE= 1 · λk or as NE·SE= ak−1 · bk, as the two are equivalent.

5.4 Laguerre Moments and the Permutation Group Sn Let us determine the weight of the fourth moment of the Laguerre polynomials for the sequence α = 0, by ﬁrst using the moment functional in the Classical approach, and second by using Viennot’s combinatorial model of colored Motzkin paths (see section 5.3). The moment functional for the Laguerre polynomials is: Z ∞ n n α −x µn = L [x ] = x x e dx = Γ(n + α + 1), (50) 0 where we see that the moments of the Laguerre polynomials can be computed from the Gamma function, Γ(x): Z ∞ Γ(x) = xt−1e−x dx. (51) 0 If t is a positive integer, then

Γ(x) = (x − 1)! (52) Looking at the fourth moment (n = 4) in the sequence α = 0:

26 Z ∞ 4 4 0 −x µ4 = L [x ] = x x e dx = 24, (53) 0 we note the following:

24 = (5 − 1)! (54) Using Viennot’s model and the bijection that exists between the Laguerre polynomials, colored lattice paths, increasing binary trees and the symmetric permutation group Sn, we can determine the moments for the same Laguerre polynomial sequence without integration. Viennot’s general result is:

µn = (α + 1)n = (α + 1)...(α + n), (55) where n is the rising factorial.

For example, the moment sequence for α = 0 is given by:

µn = n! (56) For α = 1:

µn = (n + 1)! (57) which is the particular case we explored in the preceding section (5.3). Equation (55) can also be obtained by reﬁning the model presented there, counting weighted permutations. Note also that equation (55) may be rewritten as:

Γ(n + α + 1) µ = . (58) n Γ(α + 1)

This formula diﬀers from equation (53) by a factor of Γ(α+1). As before, this is because in Viennot’s model, µ0 is always 1, whereas the Classical formula gives µ0 = Γ(α + 1).

6 Conclusion

Xavier Viennot has created several interesting and useful combinatorial models for Orthogonal Polynomial Sequences. In this paper, we have explored

27 Viennot’s Favard Paths, Motzkin Paths, Dyck Paths for the special case of symmetric Orthogonal Polynomial Sequences with symmetric intervals, permutations, binary trees, and colored lattice paths. There are still many models remaining in his work that have not been covered in this study.

References

[1] Kaye, Richard and Wilson, Robert. Linear Algebra. Oxford University Press, 1998. Print.

[2] Boas, Mary. Mathematical Methods in the Physical Sciences, Second Edition. John Wiley and Sons, Inc., 1983. Print.

[3] Chihara, Theodore. An Introduction to Orthogonal Polynomials. New York: Gordon and Breach, 1978. Print.

[4] Aharonov, D., Beardon, A., and Driver, K. “Fibonacci, Chebyshev, and Orthogonal Polynomials.” American Mathematics Monthly, 112:7 (August–September 2005) 612–630.

[5] Long, Kevin. Linear and Bilinear Functionals. Texas Tech University Department of Mathematics, 2010. Web.

[6] Abramowitz and Stegun. Classical Orthogonal Polynomials. NIST Digi- tal Library of Mathematical Functions, May 6, 2013. Web.

[7] Stanley, Richard P. Catalan Addendum. MIT Department of Mathemat- ics, May 25, 2013. Web.

[8] Stanton, Dennis. “Enumeration and Special Functions.” Lecture Notes in Mathematics 1817 (2003): 137–166. Print.

[9] Viennot, Xavier. Xavier Viennot. University of Quebec Montreal De- partment of Combinatorics and Mathematical Computer Science, 1984. Web.

[10] Sloane, Neil JA.“The On-Line Encyclopedia of Integer Sequences.” (2003).