Part 1: The Perrin Conjugate and the Laguerre Orthogonal Polynomial

3 2 3 2 I defined the conjugate of a cubic polynomial G(x) = x - Bx – Cx - D as G(x)c = x + Bx – Cx + D. By multiplying the polynomial with its’ conjugate one obtains the first polynomial of order 2;

[1] P2(x,2) = x6 + (B2-2C) x4 + (C2-2BD) x2 - D2

3 3 The Perrin sequence is associated with solutions to G(x) = x –x – 1 = 0. Its’ Perrin conjugate, G(x)c = x –x + 1 multiplied by G(x) yields a degree six polynomial,

[2] P2{x,2) = x6 -2 x4 + x2 – 1

Successive multiplication of P2(x,2) with its conjugates gives the recursive formula,

[3] P2(x,n)* P2(x,n)c = P2(x,2n).

In Part 1, I investigate another sequence which is related to the Perrin conjugate. Many well- known functions can be expanded as summations in powers of x. As an example, the exponential function ex is expanded as,

푥2 푥3 푥4 푥5 푥6 푥7 푥8 푥9 푥10 푥11 [4] F(x) = ex = 1 + 푥 + + + + + + + + + + + ⋯ .. 2 6 24 120 720 5040 40320 362880 3628800 39916800 or in terms of a summation,

푥푛 [5] F(x) = ex = ∑∞ 푛=0 푛! In this case the denominator of the expansion is a series of factorials in n. Many other exponential functions can be expressed as an expansion of factorials. For example,

푥3 푥5 푥7 푥9 푥11 푥13 [6] F(x) = Sinh[x] = 푥 + + + + + + + ⋯ .. 6 120 5040 362880 39916800 6227020800 For some functions, the denominator of the expansion does not need to be a common factorial. An x 3 example I will discuss further is the expansion of e *G(x)c, specifically when G(x)c = x –x + 1.

2 3 4 5 6 7 8 9 10 11 x 푥 2푥 7푥 7푥 23푥 17푥 47푥 31푥 79푥 7푥 [7] F(x) = e *G(x)c = 1 − + + + + + + + + + + ⋯ 2 3 8 15 144 420 5760 22680 403200 285120 In this case the denominator is not an obvious factorial and the numerator is an unknown series of coefficients. If these sequences are searched on OEIS then the numerator sequence 2,7,7,23,17,47.. is close to sequence A164314 described as the largest prime factor of (n-1)2-2. For example for n = 10, (10- 1)2-2 = 79 which is a prime and the largest factor of itself. Unfortunately, as the number of terms increases the numerator is not the largest prime factor for all terms (e.g. at n = 29, (29-1)2-2 = 782 = 23*17*2) but the factor is indicated as 1. The reason for these discrepancies will be discussed.

A search of the denominator sequence 2,3,8,15,144,420,5760… on OEIS is described as the value of the x denominator of the Laguerre polynomial of degree n at x = 1. The expansion of e *G(x)c is then related indirectly to the Laguerre polynomial.

The Laguerre Polynomial

The Laguerre polynomial is one of several classical orthogonal polynomials found in . A standard source that I will be using is found in reference (1). The closed form definition of the generalized Laguerre polynomial of degree n is

푛 + 훼 푥푖 [8] 퐿훼 (푥) = ∑푛 (−1)푖 ∗ ( ) ∗ 푛 푖=0 푛 − 푖 푖! 푛 + 훼 Where ( ) is the equal to (n+)! /((n-i)! (+i)!). 푛 − 푖 0 The Laguerre polynomial for  =0 is defined as 퐿푛(푥) = 퐿푛(푥).

The exponential for 퐿푛(1) is

푥2 2푥3 5푥4 7푥5 37푥6 17푥7 887푥8 1405푥9 168919푥10 [9] ex/x-1/(1-x) = 1 − − − − − − + + + + ⋯ .. 2 3 8 15 144 420 5760 4536 403200 Comparing equation [9] with [7] notice that some denominators are different but are integer multiples. Although this disproves that the numerator of [7] is the largest prime factor of (n-1)2-2 it strengths the hypothesis that the denominator is a denominator of 퐿푛(1). The polynomial for a given n with  =0 can be calculated from [8]. A cubic polynomial at n=3 is

1 2 [10] 퐿 (푥) = (6 − 18푥 + 9푥2 − 푥3) with 퐿 (1) = − 3 6 3 3

rd agreeing with the 3 coefficient term in [9]. Note that the coefficient for 퐿1(1) = 0 An exponential generating function can be used to find the expansion coefficients for the generalized Laguerre polynomial.

푐푥 훼−1 − 훼 [11] 퐿푛 (푐) = 푒 1−푥/(1 − 푥) Where c is an integer and  is any rational number ≥0 or ≤0.

훼−1 Equation [7] can also be generalized to give the magnitude of the denominator of 퐿푛 (푐)

훼−1 푐푥 3 훼 [12] 퐷푒푛표푚푖푛푎푡표푟[퐿푛 (푐)] = 푓 ∗ 푒 ∗ (1 − 푥 − 푥 ) where f is a positive integer generally equal to 1 and ≥0 or ≤0.

As with most orthogonal functions, any monomial term can be expressed as a sum of Laguerre polynomials. By adding these monomial terms together and multiplying by the associated coefficient any polynomial can be expressed as a sum of Laguerre polynomials.

3 1 As an example, 푥 = −6퐿3(푥) + 18퐿2(푥) − 18퐿1(푥) + 6퐿0(푥) , 푥 = −퐿1(푥) + 퐿0(푥) and

1 = 퐿0(푥) and the summation gives for the Perrin conjugate:

3 [13] 푥 − 푥 + 1 = −6퐿3(푥) + 18퐿2(푥) − 17퐿1(푥) + 6퐿0(푥) The value of the function on the LHS can then be verified for x=1 from associated Laguerre coefficients found in equation [9] or for any x = c from equation [11]

The orthogonality of the Laguerre polynomial can be useful for integrating polynomials formed from the summation of Laguerre polynomials. For =0 Laguerre polynomials satisfy the condition,

∞ −푥 [14] ∫0 퐿푛(푥) ∗ 푒 ∗ 퐿푚(푥)푑푥 = 훿푚푛 where 훿푚푛 = 1 if n=m and 0 otherwise. −푥 Multiplication of both sides of [13] by 퐿푛(푥) ∗ 푒 and integrating in the limit 0 to results in a simplification of the integration of the LHS due to orthogonality on the RHS. In general,

∞ −푥 ( ) [15] ∫0 퐿푛(푥) ∗ 푒 ∗ 퐺 푥 푑푥 = 푎푛

Where 푎푛 is the coefficient of 퐿푛(푥) for expansion of G(x) with Laguerre polynomials.

The expansion coefficients 푎푛 for each monomial can conveniently be found either in tables such as in reference (1) [Table 22.10], or by application of a Groebner basis. In Mathematica, expand the series in unknown coefficients of decreasing n using the Laguerre command and then use the Groebner basis command to solve for the coefficients. An example for obtaining x3 is,

[16] In = Expand[푎 ∗ LaguerreL[3, 푥] + 푏 ∗ LaguerreL[2, 푥] + 푐 ∗ LaguerreL[1, 푥] + 푑 ∗ LaguerreL[0, 푥]]

3푎푥2 푏푥2 푎푥3 Out = 푎 + 푏 + 푐 + 푑 − 3푎푥 − 2푏푥 − 푐푥 + + − 2 2 6

3푎 푏 In = GroebnerBasis[{ + , −3푎 − 2푏 − 푐, 푎 + 푏 + 푐 + 푑, 푎 + 6}, {푎, 푏, 푐, 푑}] 2 2 Out = {−6 + 푑, 18 + 푐, −18 + 푏, 6 + 푎}

If the expansion is made with the generalized Laguerre polynomial then the integration becomes

∞ 훼 −푥 훼 ( ) [17] ∫0 퐿푛(푥) ∗ 푒 ∗ 푥 ∗ 퐺 푥 푑푥 = 푎푛 ∗ (n +  + 1)/n! and the infinite range of integration on the LHS can be avoided by using the Gamma function of the RHS.

Expansion of Classic Orthogonal Polynomials with Generalized Laguerre Polynomials

Although a method such as [16] can be used for polynomials G(x) of any degree, as the degree increases the number of individual unknown coefficients to be solved increases as (n+1) (n+2)/2. For a polynomial of the 6th degree, 28 coefficients need to be found.

The problem posed in this section is: Can an orthogonal polynomial be expanded in terms of the generalized Laguerre polynomials? The answer is yes for the following polynomials; Legendre, Hermite, and Chebyshev. We seek a solution such that given an orthogonal polynomial Xm(x), there exists an expansion;

푋 (푥) = ∑푚 퐴 ∗ 퐿훼푖 (0) ∗ 푥푖 [18] 푚 푖=0 푚,푖 푚푖 퐴 퐿훼푖 (0) where 푚,푖 is a coefficient dependent on m and i, and 푚푖 is the generalized Laguerre polynomial evaluated at x = 0 with coefficients mi and  also dependent on i. I will use the notation of reference (1) where 푃푚(푥) is the Legendre polynomial, 퐻푚(푥) the Hermite polynomial, 푇푚(푥) the Chebyshev T polynomial, 푈푚(푥) the Chebyshev U polynomial, 푆푚(푥) the Chebyshev S polynomial, and 퐶푚(푥) the Chebyshev C polynomial.

푖 Using the orthogonality property, once these expressions are found then the nth term 푋푚푖푥 of 푃푚(푥), 퐻푚(푥), 푇푚(푥), 푈푚(푥), 푆푚(푥) or 퐶푚(푥) can be found by integration with the appropriate generalized Laguerre polynomial as in [19] and the Gamma function. ∞ 훼(푚,푖) −푧 훼(푚,푖) 훼(푚,푖) 푖 ∫ 퐿 (푧)∗푒 ∗푥 ∗푐(푚,푖)∗퐿 (푧)∗푥 푑푧 (훼(푚,푛)+푏(푚,푛)+1)∗푐(푚,푛)∗푥푖 [19] 푋 푥푖 = 0 푏(푚,푖) 푏(푚,푖) = 푚푖 (m,n)! 푏(푚,푛)!∗훼(푚,푛)! where (m,i), b(m,i) and c(m,i) are constant coefficients dependent on the degree m and the desired ith term of 푋푚(푥). In general, these integrals converge rapidly and integration to infinity is not required if performed numerically.

6 6 As an example, compute the coefficient for x (I = 6) in the degree 12 Chebyshev S polynomial, 푆12,6푥 . In equation [19] the 6th power of x is x (m-2n), n = 3 and the constants are (12,6) = n = 3, b (12,6) = m-2n = 6 and c (12,6) = (−1)푛 = -1.

∞ ∫ 퐿3(푧)∗푒−푧∗푥3∗(−1)∗퐿3(푧)∗푥6푑푧 (6+3+1)∗(−1)∗푥6 [20] 푆 푥6 = 0 6 6 = = −84푥6 12,6 3! 6!∗3!

6 The result agrees with Table 22.8 in reference (1) for matrix element A[S12(x), x ]. Since this table stops at S12(x) any coefficient of S12(x) can be determined from [20]. Note that this equation applies only for even values of m. Equations for even and odd values of m will be described for all orthogonal polynomials 푋푚(푥). Also, equation [20] converges to the value -84 to twenty decimal places for integration limits 0 to ≥100.

Associated with equation [19] which computes the individual coefficients, each orthogonal polynomial can be completely calculated using equation [18]. For the Chebyshev S polynomial, this expansion for m(even) is,

푚/2 n 푛 (푚−2푛) [21] 푆푚(푥) = ∑푛=0(−1) ∗ 퐿푚−2푛(0) ∗ 푥 2 4 6 8 10 12 [22] 푆12(푥) = 1 − 21푥 + 70푥 − 84푥 + 45푥 − 11푥 + 푥 The following set of equations [23] to [34] are expansions of each orthogonal polynomial in terms of 훼(푚,푖) 퐿푏(푚,푖)(0). They have been tested in Mathematica using the appropriate commands for the generalized Laguerre polynomial. These derived expansions may not be the most optimal in form but are the basis for the integral equation [19]. These integral equations are unique since they can be used to calculate individual terms for other orthogonal polynomials. They are important in observing symmetry in the group of monomial coefficients of orthogonal polynomials; some of these symmetries will be described later.

Orthogonal Polynomial expansions: m(even)

푚⁄2 퐻 (푥) = ∑ (−1)(푚+2푛)⁄2 ∗ (2 ∗ (푚 − 2푛)! ∗ 22푛−1⁄((푚 − 2푛)⁄2)!) ∗ L푚−2푛(0) ∗ 푥2푛 [23] 푚 푛=0 2푛

푚⁄2 푈 (푥) = ∑ (−1)n ∗ L푛 (0) ∗ (2푥)푚−2푛 [24] 푚 푛=0 푚−2푛

푚⁄2 푆 (푥) = ∑ (−1)n ∗ L푛 (0) ∗ 푥푚−2푛 [25] 푚 푛=0 푚−2푛

푚 푚 −푛 −푛−1 푚/2 (푚+2푛)⁄2 2 2 2푛 [26] 푇푚(푥) = ∑푛=0(−1) ∗ (L2푛 (0) + L2푛 (0))⁄2 ∗ (2푥)

푚 푚 −푛 −푛−1 푚/2 (푚+2푛)⁄2 2 2 2푛 [27] 퐶푚(푥) = ∑푛=0(−1) ∗ (L2푛 (0) + L2푛 (0)) ∗ 푥 (푚+2푛 −1 L 2 (0) [28] 푃 (푥) = (푑푒) ∑푚/2(−1)(푚+2푛)⁄2 (2 ∗ ((푚 + 2푛)⁄2)!)⁄(((푚 − 2푛)⁄2)! (2푛)!) ∗ (푚+2푛)/2 ∗ 푥2푛 푚 푛=0 (2∗fe)

[28a] fe = Numerator[2(푚−1)⁄(푚)!]

[28b] de = 1⁄Numerator[2푚 ∗ 푚!2⁄(2푚)!]

Orthogonal Polynomial expansions: m(odd)

(푚−1)⁄2 퐻 (푥) = ∑ (−1)(푚+2푛+3)⁄2 ∗ (2 ∗ (푚 − 2푛 − 1)! ∗ 22푛⁄((푚 − 2푛 − 1)⁄2)!) ∗ L푚−2푛−1(0) ∗ 푥2푛+1 [29] 푚 푛=0 2푛+1

(푚−1)⁄2 푈 (푥) = ∑ (−1)n ∗ L푛 (0) ∗ (2푥)푚−2푛 [30] 푚 푛=0 푚−2푛

(푚−1)⁄2 푆 (푥) = ∑ (−1)n ∗ L푛 (0) ∗ 푥푚−2푛 [31] 푚 푛=0 푚−2푛

(푚−1) (푚−1) −푛 −푛−1 (푚−1)/2 (푚+2푛−1)⁄2 2 2 2푛+1 [32] 푇푚(푥) = ∑푛=0 (−1) ∗ (L2푛+1 (0) + L2푛+1 (0))⁄2 ∗ (2푥)

(푚−1) (푚−1) −푛 −푛−1 (푚−1)/2 (푚+2푛−1)⁄2 2 2 2푛+1 [33] 퐶푚(푥) = ∑푛=0 (−1) ∗ (L2푛+1 (0) + L2푛+1 (0)) ∗ 푥

[34] 푃푚(푥) = (푚+2푛−1) L 2 (0) (푑표) ∑(푚−1)/2(−1)(푚+2푛−1)⁄2 (2 ∗ ((푚 + 2푛 + 1)⁄2)!)⁄(((푚 − 2푛 − 1)⁄2)! (2푛 + 1)!) ∗ (푚+2푛+1)/2 ∗ 푥2푛+1 푛=0 (2∗fo)

[34a] fo = Numerator[2(푚−1)⁄(푚)!]

[34b] do = 1⁄Numerator[2푚 ∗ 푚!2⁄(2푚)!]

Integral Representations of Orthogonal Polynomial Monomial terms.

∞ 훼(푚,푛) −푧 훼(푚,푛) 훼(푚,푖) 푖 ∫ 퐿푏(푚,푛)(푧) ∗ 푒 ∗ 푥 ∗ 푐(푚, 푛) ∗ 퐿푏(푚,푖)(푧) ∗ 푥 푑푧 푋 푥푖 = 0 푚푖 (m, n)! (훼(푚, 푛) + 푏(푚, 푛) + 1) ∗ 푐(푚, 푛) ∗ 푥푖 = 푏(푚, 푛)! ∗ 훼(푚, 푛)!

푖 [35] Hermite Polynomial 퐻푚푖푥 ,m even 푛 = 푖⁄2

푐(푚, 푛) = (−1)(푚+2푛)⁄2 ∗ (2 ∗ (푚 − 2푛)! ∗ 22푛−1⁄((푚 − 2푛)⁄2)!)

훼(푚, 푛) = 푚 − 2푛

푏(푚, 푛) = 2푛

( [푎 + 푏 + 1]⁄푏!) ∗ (−1)((푚+푏)⁄2) ∗ 2푏/ (푎⁄2)! ∗ 푥푏

푖 [36] Hermite Polynomial 퐻푚푖푥 ,m odd 푛 = (푖 − 1)⁄2

푐(푚, 푛) = (−1)(푚+2푛+3)⁄2 ∗ (2 ∗ (푚 − 2푛 − 1)! ∗ 22푛⁄((푚 − 2푛 − 1)⁄2)!) 훼(푚, 푛) = 푚 − 2푛 − 1

푏(푚, 푛) = 2푛 + 1

( [푎 + 푏 + 1]⁄푏!) ∗ (−1)(푎)⁄2 ∗ 2푏/(푎⁄2)! ∗ 푥푏

푖 [37] Legendre Polynomial 푃푚푖푥 ,m even 푛 = 푖⁄2

푐(푚, 푛) = (−1)(푚+2푛)⁄2 ∗ (2 ∗ ((푚 + 2푛)⁄2)!)⁄(((푚 − 2푛)⁄2)! (2푛)!) /(2 ∗ fe)

훼(푚, 푛) = (푚 + 2푛)⁄2 − 1

푏(푚, 푛) = (푚 + 2푛)⁄2

fe = Numerator[2(푚−1)⁄푚!]

(−1)푏 ∗ ( [2푏])/(((푚 − 푖)⁄2)! (푖)! ∗ (fe) ∗ 푎!) ∗ 푥푖

푖 [38] Legendre Polynomial 푃푚푖푥 ,m odd 푛 = (푖 − 1)⁄2

푐(푚, 푛) = (−1)(푚−1+2푛)⁄2 ∗ (2 ∗ ((푚 + 1 + 2푛)⁄2)!)⁄(((푚 − 1 − 2푛)⁄2)! ((2푛 + 1))!) /(2 ∗ fo)

훼(푚, 푛) = (푚 − 1 + 2푛)⁄2

푏(푚, 푛) = (푚 + 1 + 2푛)⁄2

fo = [Numerator[2(푚−1)⁄푚!]

(−1)푎 ∗ ( [2푏])/(((푚 − 푖)⁄2)! (푖)! ∗ (fo) ∗ 푎!) ∗ 푥푖

푖 [39] Chebychev S Polynomial 푆푚푖푥 ,m even 푛 = (푚 − 푖)⁄2

푐(푚, 푛) = (−1)푛

훼(푚, 푛) = 푛

푏(푚, 푛) = 푚 − 2푛

( [푎 + 푏 + 1]⁄푏!) ∗ (−1)푎 ∗ 푥푏/푎!

푖 [40] Chebychev S Polynomial 푆푚푖푥 ,m odd

푛 = (푚 − 푖)⁄2

푐(푚, 푛) = (−1)푛 훼(푚, 푛) = 푛

푏(푚, 푛) = 푚 − 2푛

( [푎 + 푏 + 1]⁄푏!) ∗ (−1)푎 ∗ 푥푏/푎!

푖 [41] Chebychev U Polynomial 푈푚푖푥 ,m even 푛 = (푚 − 푖)⁄2

푐(푚, 푛) = (−1)푛 ∗ 2(푚−2푛)

훼(푚, 푛) = 푛

푏(푚, 푛) = 푚 − 2푛

( [푎 + 푏 + 1]⁄푏!) ∗ (−1)푎 ∗ 2푏 ∗ 푥푏/푎!

푖 [42] Chebychev U Polynomial 푈푚푖푥 ,m odd 푛 = (푚 − 푖)⁄2

푐(푚, 푛) = (−1)푛 ∗ 2(푚−2푛)

훼(푚, 푛) = 푛

푏(푚, 푛) = 푚 − 2푛

( [푎 + 푏 + 1]⁄푏!) ∗ (−1)푎 ∗ 2푏 ∗ 푥푏/푎!

The following Chebyshev polynomials consist of two integrations

∞ 훼(푚,푛) −푧 훼(푚,푛) 훼(푚,푖) 푖 ∫ 퐿푏(푚,푛)(푧) ∗ 푒 ∗ 푥 ∗ 푐(푚, 푛) ∗ 퐿푏(푚,푖)(푧) ∗ 푥 푑푧 푋 푥푖 = 0 푚푖 (m, n)! ∞ 훼1(푚,푛) −푧 훼1(푚,푛) 훼1(푚,푖) 푖 ∫ 퐿푏(푚,푛) (푧) ∗ 푒 ∗ 푥 ∗ 푐(푚, 푛) ∗ 퐿푏(푚,푖) (푧) ∗ 푥 푑푧 + 0 1(m, n)!

(훼(푚,푛)+푏(푚,푛)+1)∗푐(푚,푛)∗푥푖 푋 푥푖 = + 푚푖 푏(푚,푛)!∗훼(푚,푛)! (훼1(푚,푛)+푏(푚,푛)+1)∗푐(푚,푛)∗푥푖

푏(푚,푛)!∗훼1(푚,푛)!

푖 [43] Chebychev T Polynomial 푇푚푖푥 ,m even 푛 = 푖⁄2 푐(푚, 푛) = (−1)(푚+2푛) ∗ 22푛/2

훼(푚, 푛) = 푚⁄2 − 푛 − 1

푏(푚, 푛) = 2푛

훼1(푚, 푛) = 푚⁄2 − 푛

( [푎 + 푏 + 1]⁄푏!) ∗ (−1)((푚+푏)⁄2) ∗ 2(푏−1) ∗ 푥푏⁄푎! + ( [푎1 + 푏 + 1]⁄푏!) ∗ (−1)((푚+푏)⁄2) ∗ 2(푏−1) ∗ 푥푏⁄a1!

푖 [44] Chebychev T Polynomial 푇푚푖푥 ,m odd 푛 = (푖 − 1)⁄2

푐(푚, 푛) = (−1)(푚+2푛−1) ∗ 22푛+1/2

훼(푚, 푛) = (푚 − 1)⁄2 − 푛 − 1

푏(푚, 푛) = 2푛 + 1

훼1(푚, 푛) = (푚 − 1)⁄2 − 푛

( [푎 + 푏 + 1]⁄푏!) ∗ (−1)((푚+푏−2)⁄2) ∗ 2(푏−1) ∗ 푥푏⁄푎! + ( [푎1 + 푏 + 1]⁄푏!) ∗ (−1)((푚+푏−2)⁄2) ∗ 2(푏−1) ∗ 푥푏⁄a1!

푖 [45] Chebychev C Polynomial 퐶푚푖푥 ,m even 푛 = 푖⁄2

푐(푚, 푛) = (−1)(푚+2푛)⁄2

훼(푚, 푛) = 푚⁄2 − 푛 − 1

푏(푚, 푛) = 2푛

훼1(푚, 푛) = 푚⁄2 − 푛

( [푎 + 푏 + 1]⁄푏!) ∗ (−1)((푚+푏)⁄2) ∗ 2(푏) ∗ 푥푏⁄푎! + ( [푎1 + 푏 + 1]⁄푏!) ∗ (−1)((푚+푏)⁄2) ∗ 2(푏) ∗ 푥푏⁄a1!

푖 [46] Chebychev C Polynomial 퐶푚푖푥 ,m odd 푛 = (푖 − 1)⁄2

푐(푚, 푛) = (−1)(푚+2푛−1)⁄2

훼(푚, 푛) = (푚 − 1)⁄2 − 푛 − 1

푏(푚, 푛) = 2푛

훼1(푚, 푛) = (푚 − 1)⁄2 − 푛

( [푎 + 푏 + 1]⁄푏!) ∗ (−1)((푚+푏−1)⁄2) ∗ 2(푏) ∗ 푥푏⁄푎! + ( [푎1 + 푏 + 1]⁄푏!) ∗ (−1)((푚+푏−1)⁄2) ∗ 2(푏) ∗ 푥푏⁄a1!

The following computations are easily done with Mathematica. Integration limits are typically 10*m. Calculation with the Gamma function is faster.

17 Compute 푃23,17푥 (use integration limit 0 to 10*23)

fo = [Numerator[2(23−1)⁄23!]= 8 n = (17-1)/2 = 8

훼(푚, 푛) = (23 − 1 + 2(8))⁄2 = 19 푏(푚, 푛) = (23 + 1 + 2(8))⁄2 = 20

(2∗(20)!)⁄(3)!(17)! 푐(푚, 푛) = (−1)19 ∗ = −2280/16 2∗8

10∗푚 19 −푧 19 2280 19 17 ∫ 퐿20(푧) ∗ 푒 ∗ 푥 ∗ (−1) ∗ ∗ 퐿20(푧) ∗ 푥 푑푧 39! 푃 푥17 = 0 16 = − 23,17 19! 6 ∗ 17! ∗ 8 ∗ 19! = −9821565178425푥17

This number agrees with the expansion for 푃23(푥) found in equation [34]

3 5 7 9 푃23(푥) = −2028117푥 + 185910725푥 − 5019589575푥 + 62386327575푥 − 429772478850푥 + 1805044411170푥11 − 4859734953150푥13 + 8562390155550푥15 − 9821565178425푥17 + 7064634602025푥19 − 2893136075115푥21 + 514589420475푥23

Equations [43] and [44] and [45] and [46] demonstrate that

푖 푖 푖 푇푚푖푥 = 1/2(푈푚푖푥 − 푈(푚−2)푖푥 )

푖 푖 푖 퐶푚푖푥 = (푆푚푖푥 − 푆(푚−2)푖푥 ) This symmetry is also listed in [22.5.8] and [22.5.12] of reference (1).

Monomial Term Symmetry

Using the above integral representations, knowing conditions for symmetry in the generalized Laguerre polynomial leads to symmetry of one or more of the orthogonal polynomials. As a simple example, 푖 consider the Legendre polynomial 푃푚,푖푥 . For any m, if i = m then given m1 = 2m and i = 0 the 푚 0 coefficients for the generalized Laguerre polynomial are equal and 푃푚,푚푥 = −푃푚,0푥 . Also, looking 푚 푘 푚−푘 0 0 k at the Hermite polynomial 퐻푚,푚푥 = 2 퐻푚−푘,푚−푘푥 where 퐿푚,푚 and 퐿푚−푘,푚−푘 differ by 2 .

푓(푚) For the Legendre polynomial, the relation 푃 푥푚−2 = ∗ 푃 푥1 depends on the 푚,푚−2 푓(2(푚−2)+1) 2(푚−2)+1,1 values of f(m) = fo(m) or fe(m) with m even or odd. The Legendre polynomial equations [37] and [38] are suitable for finding several symmetries. For example, by partitioning (a+b) in [37 and [38] into 3 parts the following relation is found,

푖 푓(푚/2 + 3푖/2) (푚−푖)/2 푃 푥 = ∗ 푃푚 3푖 푥 푚,푖 + ,(푚−푖)/2 푓(푚) 2 2 Unfortunately, the generalized Laguerre polynomial has no known recurrences relating 훼(푚, 푛) and 푏(푚, 푛) to another different set 훼(푚, 푛) and 푏(푚, 푛) with 훼(푚, 푛)>0. However, the Laguerre polynomial is directly related to a confluent hypergeometric function, the Kummer function M[a, b, z] (see reference (2), equation 10.12.14]. A direct substitution of the Kummer polynomial function for the Laguerre polynomial in the expansion of the Hermite polynomial [23] yields;

푚⁄2 푚! 퐻 (푥) = ∑(−1)(푚+2푛)⁄2 ∗ (2 ∗ (푚 − 2푛)! ∗ 22푛−1⁄((푚 − 2푛)⁄2)!) ∗ ( ∗ M[−2n, m − 2n + 1,0]) ∗ 푥2푛 푚 (푚 − 2푛)! ∗ (2푛)! 푛=0 The integral representation of the monomials for the Hermite polynomial in terms of the Kummer function from [35] is:

∞ 푖 훼(푚,푛) −푧 훼(푚,푛) 푚! 푖 퐻푚푖푥 = (∫ 퐿푏(푚,푛)(푧) ∗ 푒 ∗ 푥 ∗ 푐(푚, 푛) ∗ ( ∗ M[−푏(푚, 푛), 훼(푚, 푛) + 1, z]) ∗ 푥 푑푧)/(m, n)! 0 훼(푚, 푛)! ∗ 푏(푚, 푛)!

훼(푚,푛) where the Kummer term in parenthesis can also replace 퐿푏(푚,푛)(푧). Further research in this area is required to discover hidden symmetries using other representations of the confluent hypergeometric functions.

The above equations, [35] to [46] illustrate a direct connection of the orthogonal polynomials to symmetric functions. The term ( [푎 + 푏 + 1]⁄푎! 푏!) = (푎 + 푏, 푏)! is the integer coefficient of the term a b (a+b) x1 x2 in the expansion of (x1+x2) . For the Chebyshev S and U polynomials,

b a b The x term of the Chebyshev polynomial Xmb is c(m,n) *[coefficient of x1 x2 ] in the expansion of (a+b) (x1+x2) .

A similar connection can be made for the Hermite and Legendre polynomials:

b a b The x term of the Hermite polynomial Hmb is c(m,n) *[coefficient of x1 x2 ] in the expansion of (a+b) (x1+x2) .

i a b The x term of the Legendre polynomial Pmi is c(m,n) *[coefficient of x1 x2 ] in the expansion of (a+b) (x1+x2) .

a b b a These functions are symmetric since the coefficients x1 x2 = x1 x2 . The relation of univariate orthogonal polynomials to bivariate symmetric polynomials has many applications in algebraic number theory particularly in symmetric group representation and combinatorial algorithms. See reference (3).

This study began with investigation of equation [7]. The sequence in the denominator is found equivalent to the exponential generating function for the Laguerre polynomial evaluated at unity [9]. The Perrin conjugate equation multiplies the exponential and shifts the factorial denominator in powers of x. For this reason, the denominator of [7] follows a series 1/n! -1/(n-1)! + 1/(n-3)!. This result is equivalent to the nth term found in [7]. Although equation [7] did not lead to any further mathematical consequence, the concept lead to the use of the generalized Laguerre polynomial in the expansion of other classical orthogonal polynomials as demonstrated in equations [23] to [46]. It is hoped that these expansions will be a useful reference to those using classical orthogonal polynomials.

(1) Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series 55, M. Abramowitz and I. Stegun (Ed), (1964) pp 771-802. (2) Higher Transcendental Functions, Vol. II, Bateman Manuscript Project, H. Bateman, (1953), pp 178-196. (3) The Symmetric Group [Ed. 2], B. Sagan, Springer, (2000).

Part 2: The Jacobi Polynomial, Laguerre Polynomial and Delannoy Number

In Part 1 we found Laguerre expansions of various orthogonal polynomials. Since the Laguerre polynomial L(m, a, x) is also orthogonal, these expansions could be integrated over the positive x axis with weight e-x * xa to give the coefficient of each term of degree xn. This was done for various Chebyshev polynomials, the Hermite polynomial and the Legendre polynomial.

The Jacobi Polynomial P(m, a, b, x) is a general orthogonal form of the Legendre Polynomial P(m, x). For integers a = 0 and b= 0, P(m, 0, 0, x) = P(m, x). The Jacobi polynomial is given as an expansion in terms of (x - 1)n so the complete form in xn requires an expansion in powers of (x - 1)n. It is convenient to start with

[1]

From the definition of the Laguerre Polynomial it is simple to show that this can be written as:

[2]

Integrating the term in the summation of [2] wrt variable z and L(b+m, a+n, z),

[3] we obtain,

[4]

When the terms preceding the summation in [2] are included, the nth coefficient in (x - 1)n is

[5]

Since the Legendre and Jacobi polynomials are usually given as coefficients in xn, it is necessary to expand and collect all terms (x - 1)n . This is best done by the summation of [2];

[6]

In Mathematica, the function [6] can be directly expanded to yield terms in xn.

It is interesting to compare equations [2] to [6] with the equations obtained for the Legendre polynomial in Part 1. It can be shown that equation [28] and [34] in that Part give the same result as equation [6] for P(m, 0, 0, x) as expected. However, since the maximum degree of P(m, 0, 0, x) is m, there is only one xm term to collect in [6]. In equation [5] let a = 0, b = 0 and n = m. Then the final term xm, in [6] is given simply by, [7]

Unexpectedly, the denominator of this term is the constant de and do defined previously

[8] and removing the Numerator designation,

[9]

Also, the terms fe and fo previously described for computing the Legendre polynomial were found in OEIS A048896. By simplifying the terms for xm in equation [28] and [34] of the previous Part, the following equations are derived,

[10a, b] fe =

fo =

The orthogonality of P(m, a, b, x) can be proved by integrating [2] wrt z with L(b + m’, a + n, z) where m’ ≠ m for all terms in n,

[11]

After expanding equation [6] standardized orthogonal polynomials are obtained with [12] as standardization,

[12]

such that

[13] if m = m’ and zero, otherwise.

Also, equations [1] and [2] are solvable with positive and negative integers and half fraction values for parameters a and b.

The Delannoy Number

The Delannoy number D(m, n) is an integer sequence which quantifies the number of walks on a lattice plane from point (0, 0) to point (m, n). The constraint is that only walks (0, 1), (1, 0) and (1, 1) are allowed and they can only occur in a north, east or northeast direction. Given a central point (m, m) on a square lattice the value of D(m, m) = P(m,3) the value of the Legendre polynomial at x = 3. For example D(2, 2) = P(2, 3) = 13. An illustration of these walks from (0, 0) to (2, 2) is shown:

There is no known combinatoric correspondence of the Legendre Polynomial to the Delannoy numbers (see below Delannoy numbers revisited). However, Gabor Hetyei (reference (1) demonstrated that the asymmetric Delannoy number could be expressed as a Jacobi Polynomial P(n, 0, m-n, 3). is the number of walks on a lattice plane from (0, 0) to (m, n + 1) where each step (x,y) is a nonnegative integer in x and a positive integer in y. Expressing a = 0, b = m – n and x = 3 in equation [5] the following equation for the asymmetric Delannoy number is derived: (note: the variable m is exchanged with n and the original n is replaced with the summation variable k)

[14]

As an example of [14] with n = 3 and m= 2, and b = m - n = -1, = 38. (see Table of Asymmetric Delannoy Numbers in reference (1)).

A more extensive table read as diagonals of the table in reference (1) is given in OEIS A049600. The (m+n)th diagonal is calculated by the following equation using Laguerre polynomials derived from [2];

[15]

To use this equation n’ = m + n and n = n. From the example above n’ = 5 and n = 3 giving the same result = 38. (See OEIS A049600 for more detail).

Equation [15] also has a connection to n-simplices. An n-simplex is an n dimensional triangle. Starting with a triangle 2-simplex of 3 vertices adding a point orthogonal to the other 3 points represents a tetrahedron. The tetrahedron has 4 vertices, 6 edges, 4, 2-faces and 1, 3- face which is the whole 3-simplex tetrahedron. Adding another point to the 3-simplex results in a 4-simplex or 5-cell of 5 vertices, 10 edges, 10, 2 faces, 5, 3 faces and 1, 4- face.

It can be shown that if i = (i - 1)th-face, and n’ = ( n - 1)-Simplex then integrating [15] with the Laguerre polynomial yields,

[16]

Further setting n = i results in a simple binomial equation i] for the number of (i-1)-faces for the n’ = (n-1)-Simplex. For example, n’ = 8 represents the 7-Simplex and i = 5 is the 4-face then;

[17] Number of 4-faces of a 7-simplex = = 56

The f-vector for an n-simplex is the series of faces {f-1, f0, f1, ….., f(n-1)} where f-1=1 and the series represents the number of points, edges, 2-face, 3-face etc. The associated f- polynomial uses this vector as coefficients, n n-1 n-2 [18] f(x) = f(-1), *(x - 1) + f0 *(x - 1) + f1 * (x - 1) +……..+ f(n-1)

The expansion of [18] results in the corresponding polynomial h(x) with coefficients for powers of x.

n n-1 n-2 [19] h(x) = h(-1), *(x ) + h0 *(x ) + h1 * (x ) +……..+ h(n-1) where the h vector is {h0, h1,…..h(n-1)} calculated by equation [20],

[20]

For all of the n=simplices the h vector is a series of n, 1’s; {1, 1, 1, 1, ……}

Hetyei discusses a 2-color interpretation of the asymmetric Delannoy number and its relation to the Jacobi and Legendre polynomial. He proves that enumerates all 2-colored lattice paths from points (0, 0) to (m, n), m and n are nonnegative integers, under the following rules:

(i) Each step is either a north direction blue (0, 1) or a red (x, y) with x and y nonnegative integers not both zero. (ii) At least one of any two consecutive steps is blue (0, 1).

He shows that the number of 2- colored lattice paths is;

[21] = where j is the number of blue steps. From [21] we see that the number of blue steps ranges from 0 to n. The steps must have a partial order such that (p,q) < (p’, q’) iff p ≤ p’ and q < q’. As an example, consider paths from (0, 0) to (1, 3). If we call the number of blue steps the “partial chains” then the number of partial chains containing 0, 1, 2, and 3 blue steps is calculated from [21] is (1, 6, 9, 4). The sum is . The first chain has no blue but only a red chain from (0, 0) to (x, y) = (1,3). To identify the remaining chains, define the following 6-line segments: a = (0, 0) to (0, 1), b = (0, 1) to (0, 2), c = (0, 2) to (0,3), d = (1, 0) to (1, 1), e = (1, 1) to (1, 2) and f = (1, 2) to (1, 3). The 6-one chains are then (a, b, c, d, e, f) the 9-two chains are (ab, bc, a and c, de, ef, d and f, a and e, a and f, b and f) and the 4-three chains (ab and f, a and ef, abc, def). The lattice paths are then completed by inserting red steps in the remaining space.

Hetyei compares these partial chains to simplicial complexes which are a set of vertices and line segments (the blue steps). Comparing the partial chains to the f -vector, this f- vector can be expanded in powers of (x-1)/2 to yield a corresponding Jacobi polynomial. In the example above {f-1, f0, f1, f2} = {1, 6, 9, 4). Then we find,

[22]

Apply equation [22] with x = 3 to give the correct result .

If we dissect equation [21] into the individual paths for a given j, each term is the product . Comparing to equation [16] above the first binomial is the number of (j-1) faces for the (n-1) simplex. Each term in the f-vector is then a (j-1)-face of the (n-1) simplex times a binomial which depends on j and m. The above example of the f vector {f-1, f0, f1, f2} = {1, 6, 9, 4) is generated from [21] with n= 3, m= 1 and j= 0 …n. This can be designated as an (n-1) simplex vector (1, s0, s1,..sn) representing a 2-simplex (triangle) with {j-1} = (-1, 0, 1, 2) terms; 1, 3-vertices, 3 edges and 1-triangle

(the (-1) term is always 1). Each j-1 term is multiplied by a “property” vector (1, p0, p1.., pn), , for the (j-1)-face.

Conjecture 1: The asymmetric Delannoy number D~ (m,n) is the dot product of the (n-1) simplex vector s with the (j-1) property vector p.

Example: Let (m,n) = (2,9). The 8-simplex has the s vector {1,9,36,84,126,126,84,36,9,1}. The term can also be equivalently expressed as . For m = 2 the p vector is {1,3,6,10,15,21,28,36,45,55}. The vector product results in the sum of terms in the f-vector {1,

27, 216, 840, 1890, 2646, 2352, 1296, 405, 55} = 9728. Looking at the s0 = 9 vertices of the 8-simplex if we append or join each of the p0 = 3 distinct ‘properties’ to each vertex we obtain 27 distinct possible outcomes. To accomplish this requires making 3 copies of the 0-face of the 8-simplex and attaching one distinct property to each of the nine vertices. The process is repeated for each (j-1) face of the 8-simplex until the 8-simplex is replicated 55 times and each receives a distinct property of p9.

The property term can be interpreted as the number of distinct colorings of k objects with c colors using all colors at least once. The binomial provides an exact interpretation by letting m+1 = number of colors and m+1+j be the number of objects to be colored. In the example j = 1 ((j-1) is the 0-face) so we have 4 objects and 3 colors. Let a, b, c, represent the three colors then there are 3 distinct colorings abca, abcb, and abcc. Apply each coloring to copies of the 9 – 0 faces. Note that the size of the objects increases by one as j increases, until j = 9 where the appended property consists of 12 objects with 55 distinct colorings using 3 colors.

The asymmetric Delannoy number is then the total number of these complexes appended to multiple copies of the (n-1) simplex. The total number of copies is the sum of the terms in the p-vector and each copy has a unique property.

Hetyei extends his theory to simplicial complexes of higher dimension and shows that the asymmetric Delannoy number counts the number of partial chains in a “balanced (all vertices colored differently) join” of simplicial complexes with an (n-1) dimensional colored simplex. Details can be found in reference (2).

The property vector can also be defined from the symmetry group, Sm. Given (m,n), the (n-1) simplex vector is defined by the faces of the (n-1) simplex. The property vector is generated from the cycle index of the symmetry group Sm. The cycle index polynomial for S2 is . If x1 and x2 are indexed as the value of m+j-1 (j = 0 …. n) the values of the cycle index polynomial are {1,3,6,10,15,21,28,36,45,55} for n

= 9 as above. The symmetry group Sm is used as the property vector for all values of m to calculate the asymmetric Delannoy number. It would be interesting to define numbers using a different finite group G to define this property such as the cyclic group or the dihedral group. The interpretation of the resulting number is left as an exercise. The Delannoy number revisited Contrary to the literature the Delannoy number D(m,n) has a close connection to the Jacobi polynomial beyond the central diagonal. The following connection follows the same proof as above. Expressing b = 0, a = m – n and x = 3 in equation [5] the following equation for the Delannoy number is derived: (note: the variable m is exchanged with n and the original n is replaced with the summation variable k):

[23] As an example of [14] with n = 3 and m= 2, and a = m - n = -1, = 25. (see Table of Delannoy Numbers in reference (1)).

The Delannoy number is also calculated by the following equation using Laguerre polynomials derived from [2] (compare with [15] above;

[24]

It is not surprising that Equation [24] also has a connection to n-simplices. Equation [24] can be written in binomial form:

[21] =

Note the slight difference in the second binomial term from equation [21]. As expected there is a similar correlation of the Delannoy number to the (n-1) simplex and a property vector as we found for the asymmetric Delannoy number. The following conjecture defines this correlation. Conjecture 2: The Delannoy number D (m,n) is the dot product of the (n-1) simplex vector s with an n- property vector p. Base on the previous analysis for the asymmetric Delannoy number the most convenient property vector is from a shifted cycle index polynomial for the symmetric group Sn. The shift is positive for m > n, negative for m < n, and no shift for m = n.

Examples: Calculate D(m,n) = D(3,6). The (n-1) simplex vector s of the f-1 = 1, and 0- face to the 5-face is

{1, 6, 15, 20, 15, 6, 1}. Consecutive values for the cycle index polynomial of the symmetric group S6 are 1, 7, 28, 84, 210, 482, …. The p vector is shifted left by 3-6 = -3 units by replacement with zeros. The remaining n+1-3 = 4 places in the vector are filled as {0, 0, 0, 1, 7, 28, 84}. The dot product of these vectors is 377. For the reverse, D(m,n) = (6,3) the corresponding 2-simplex vector is multiplied by a right shifted by 3 vector of S3. {1, 3, 3, 1} . {20, 35, 56, 84} = 377. This example demonstrates a symmetry in the Delannoy number: D(m,n) = D(n,m). For the central numbers, the dot product of a (n-1) simplex vector with un-shifted values for the cycle index polynomial of the symmetric group Sn. results in the value of the n-Legendre polynomial in x evaluated at x = 3.

Conjecture 3: The Delannoy number D(m,n) is the dot product of the n-th row of Pascal’s triangle with the m-th extended diagonal of the coefficients for the Chebyshev U polynomial.

Chebyshev U polynomials were developed from an expansion of the Laguerre polynomial in Part 1. It was shown in equation [42] that the monomials of the Chebyshev U polynomial can be expressed by:

[22]

Removing the sign and the variable x the up-diagonal coefficients can be expressed as:

[23] with b increasing from 0 to m and a decreasing as m-b. The m-th diagonal has m+1 terms but is extended by adding zeros to the end of the vector. For example. When m = 4 the first 5 terms are extended with zeros as {1, 8, 24, 32, 16, 0, 0, 0, 0, 0, 0…). The 4-th row D(4, n) is then developed by taking the dot product with the n-th row of Pascal’s triangle {the (n-1) simplices are a subset of the Pascal triangle}. Then D(4, 7) = {1, 7, 21, 35, 35, 21, 7, 1}.{{1, 8, 24, 32, 16, 0, 0,0} = P(4, 7-4, 0, 3) = 2241.

OEIS A001846 describes the D(4, n) sequence of as the “crystal ball sequence for a 4-dimensional cubic lattice”. Looking at other sequences, D(3, n) is OEIS A001845, the “crystal ball sequence for a cubic (3- dimensional) lattice” or centered octahedral numbers. D(2, n) is OEIS A001844, the “crystal ball sequence for a square (2-dimensional) lattice”. D(5, n) is OEIS A001847, the “crystal ball sequence for a 5- dimensional cubic lattice”. OEIS (reference 3) defines the crystal ball sequence as the n-th term giving the number of vertices which are at most n edge traversals away for a given vertex on a vertex transitive graph [of dimension m].

These results confirm that there is an underlying topological significance of the Delannoy and asymmetric Delannoy numbers. Both numbers are also expressible with Jacobi polynomials, Laguerre polynomials and binomials. 1. G. Hetyei, Links we almost missed between Delannoy numbers and Legendre polynomials 2. G. Hetyei, Central Delannoy numbers, Legendre polynomials, and a balanced join operation preserving the Cohen- Macaulay property, Formal Power Series and Algebraic Combinatorics, Series Formelles et Combinatoire Algebrique, San Diego, CA., 2006. 3. J. H. Conway, N. J. A. Sloane, Low-Dimensional Lattices. VII. Coordination sequences, Proceedings of the Royal Society, 453:1966 (1997), pp. 2369-2389.

Part 3: The Jacobi Polynomial Revisited

Discoveries can be fortuitous if we look in the right places. Look back at Part 2 and discover some of the properties of the Jacobi Polynomial. We saw how the combinatoric Delannoy numbers can be written as Jacobi Polynomials evaluated at x = 3. We also showed how these values can be evaluated from vector dot products of rows of the Pascal triangle with either the appropriate cycle index of a symmetry group polynomial or an appropriate diagonal of the Chebyshev U polynomials.

I give a proof in this part that all Jacobi Polynomials P[m, a, b, x] evaluated at x=3 can be reduced to a series of Jacobi Polynomials of form P[n,c,1,3] with n= 0, 1, 2, … m. Each polynomial of the series is a vector dot or scalar product of rows of the Pascal triangle with an appropriate diagonal of the Chebyshev T polynomial. Any scalar product with the Chebyshev T polynomial can be reduced to a scalar product with a shifted cycle index polynomial of a symmetric group. The result proves that the scalar value of all Jacobi polynomials P[m, a, b, 3] can be transformed to scalar products. Some Preliminaries

Three examples of the scalar product used to calculate P[m, 0, b, 3] and P[m, a, 0, 3] are reviewed from Part 2. Rows of the Pascal triangle are used in all three calculations.

Let m = 7, a=0, and b=5. Write P[7, 0, 5, 3] as P[7, 0, 12-7, 3]. Then using the (m-1)th row of Pascal, and the cycle index of the symmetry group S12, apply the scalar product to obtain;

[1] {1,7,21,35,35,21,7,1}. {1,13,91,455,1820,6188,18564,50388} = 391912 Let m = 7, a=5, and b=0. Write P[7, 5, 0, 3] as P[7, 12-7, 0, 3]. Then using the (m+a-1)th row of Pascal, and the diagonal of the Chebyshev U polynomial for xm, apply the scalar product as described in Part 2 to obtain;

[2] {1,12,66,220,495,792,924,792,495,220,66,12,1}. {1,14,84,280,560,672,448,128,0,0,0,0,0} = 1392065

Note that the ith entry of the diagonal is given by the binomial equation [27] of Part 2. As the diagonal only has 7 terms the remaining elements of the vector are assigned zero value.

It is also possible to write P[7, 5, 0, 3] as P[7, 7-2, 0, 3] and use the shifted cycle index of the symmetry th group S7. Using the 6 row of the Pascal triangle and shifting the cycle index 5 units to the right:

[3] {1,7,21,35,35,21,7,1}. {792,1716,3432,6435,11440,19448,31824,50388} = 1392065

Not that the Legendre polynomial P[7,3] with a= b= 0 can be calculated when the cycle index of S7 is not shifted.

[3a] {1,7,21,35,35,21,7,1}. {1,8,36,120,330,792,1716,3432} = 48639

The above examples illustrate that any Legendre or Jacobi polynomial can be calculated if either or both a and b are zero. Below I discuss the general Jacobi Polynomial with a and b not equal to zero and consider whether a scalar product can be used to calculate P[m, a, b, 3]a . Reduction of the Jacobi Polynomial

I define the reduction of a general Jacobi polynomial as a transform of the value of the polynomial |P[m, a, b, 3]| to a summation of the form,

푖=푚−1 [4] |P[푚, 푎, 푏, 3]| = |∑푖=0 푐푖 ∗ P[푚 − 푖, 푎 + 푖, 1,3]| + 푐푚 This formula is valid only for the scalar value of the polynomial at x = 3 and is not applicable to all values of x. As shown below it is only valid at x = 3 and for real and complex values to a difference polynomial of degree m-1.

Several identities lead to equation [4]. Consider a simple polynomial of the form P[2, 2, i, 3] with i any positive integer. Let I = 4, then it can be shown that the value of the polynomial at x = 3 is;

푖=13 [5] 푃[2,2,4,3] = P[2,2,1,3] + ∑푖=11 푖 where the summations are based on the sums m+a+i+3*2-1. This equation only applies for m = 2 and a = 2. However, it leads to an interesting and important identity. Evaluate P[3,1,4,3]:

푖=4 P[3,1,4,3] = P[3,1,1,3] + ∑ P[2,2, 푖, 3] [6] 푖=2 Using [5] the summation is expanded as

푖=11 푖=12 푖=13 [7] P[3,1,1,3] + 3 ∗ P[2,2,1,3] + ∑푖=11 푖 + ∑푖=11 푖 + ∑푖=11 푖 The summations can be combined using OEIS A101860 to give the value of P[3, 1, i, 3] for any i;

[8] P[3,1,1,3] + (푖 − 1) ∗ [2,2,1,3] + (3 + (푖 − 2)) ∗ (2 + 33(푖 − 2) + (푖 − 2)^2)⁄6 + 10

A similar formula can be used to evaluate P[3, 2, i, 3]; [9] P[3,2,1,3] + (푖 − 1) ∗ P[2,3,1,3] + (3 + (푖 − 2)) ∗ (2 + 33(푖 − 2) + (푖 − 2)^2)⁄6 + (푖 − 1) ∗ (푖) + 10

Let P[m, a, i, 3] be a general Jacobi polynomial with x = 3. Then the following identities are true.

푖=푚 푃[푚, 푎, 푖, 3] = 푃[푚, 푎, 1,3] + ∑ 푃[푚 − 1, 푎 + 1, 푖, 3] [10a] 푖=2

푖=푚 푖=푚 푃[푚, 푎, 푏, 3] = ∑ 푃[푖, 푚 + 푎 − 푖, 푏 − 1,3] = ∑ 푃[푚 − 푖, 푎 + 푖, 푏 − 1,3] [10b] 푖=0 푖=0 푖=푏 [10c] |P[푚, 푎, 푏, 3]| = |P[푚, 푎, 1,3] + ∑푖=2 P[푚 − 1, 푎 + 1, 푖, 3] | Using identities [10a] to [10c] it is possible to reduce any Jacobi polynomial at x = 3 to a form shown in equation [4].

______

푖=4−1 Example: Reduce P[4,3,3,3] (value is 4809) to |∑푖=0 푐푖 ∗ P[4 − 푖, 3 + 푖, 1,3]| + 푐4 Using [10c] and then [10a] ;

푖=3 P[4,3,3,3] = P[4,3,1,3] + ∑ P[3,4, 푖, 3] [11] 푖=2

푖=3 = P[4,3,1,3] + P[3,4,2,3] + ∑ P[3,4, 푖, 3] [11b] 푖=3

푖=3 = P[4,3,1,3] + P[3,4,1,3] + P[2,5,2,3] + ∑ P[3,4, 푖, 3] [11c] 푖=3

푖=3 = P[4,3,1,3] + P[3,4,1,3] + P[2,5,1,3] + P[1,6,2,3] + ∑ P[3,4, 푖, 3] [11d] 푖=3

푖=3 = P[4,3,1,3] + P[3,4,1,3] + P[2,5,1,3] + P[1,6,1,3] + P[0,7,1,3] + ∑ P[3,4, 푖, 3] [11e] 푖=3

푖=4 = ∑ P[4 − 푖, 3 + 푖, 1,3] + P[3,4,3,3] [11f] 푖=0 Reduce P[3,4,3,3] starting again with [10c] ;

푖=3 P[3,4,3,3] = P[3,4,1,3] + ∑ P[2,5, 푖, 3] [12a] 푖=2

푖=3 = P[3,4,1,3] + P[3,4,2,3] + ∑ P[2,5, 푖, 3] [12b] 푖=3

푖=2 = P[3,4,1,3] + ∑ P[2 − 푖, 5 + 푖, 1,3] + P[2,5,3,3] [12c] 푖=0

푖=2 푖=2 = P[3,4,1,3] + ∑ P[2 − 푖, 5 + 푖, 1,3] + ∑ P[2 − 푖, 5 + 푖, 2,3] [12d] 푖=0 푖=0

푖=2 = P[3,4,1,3] + ∑ P[2 − 푖, 5 + 푖, 1,3] + P[2,5,2,3] + P[1,6,2,3] + P[0,7,2,3] [12e] 푖=0

푖=2 푖=2 푖=1 푖=0 = P[3,4,1,3] + ∑ P[2 − 푖, 5 + 푖, 1,3] + ∑ P[2 − 푖, 5 + 푖, 1,3] + ∑ P[1 − 푖, 6 + 푖, 1,3] + ∑ P[0 − 푖, 7 + 푖, 1,3] [12f] 푖=0 푖=0 푖=0 푖=0 Combining [11f] and [12f], collecting similar terms and noting that P[0, a, 1, 3] = 1 the reduced form is;

푖=3 [13] P[4,3,3,3] = P[4,3,1,3] + 2 ∗ P[3,4,1,3] + 3 ∗ P[2,5,1,3] + 4 ∗ P[1,6,1,3] + 5 = ∑푖=0 푐푖 ∗ P[4 − 푖, 3 + 푖, 1,3] + 푐푚 Computer programming of the steps allows the lengthy calculations to be done efficiently.

Comparing the Jacobi polynomial with the reduced polynomial we find that they do not represent the same polynomial although they have the same degree m. 7 P[3,4,3, 푥] = (3 − 66x2 + 143x4) [14a] 16

3 (79 + 198x + 300x2 + 330x3 + 165x4) [14b] Reduced polynomial(x) = 16 Factoring the difference of the two polynomial we find the difference to be zero at x = 3 as expected;

1 P[3,4,3, 푥] − 푟푒푑푢푐푒푑 푝표푙푦푛표푚푖푎푙(푥) = (−3 + 푥)(36 + 111푥 + 264푥2 + 253푥3) [15] 8

If x is a solution to the degree 3 polynomial in [15] we find that |P[3,4,3, 푥]| = |푟푒푑푢푐푒푑 푝표푙푦푛표푚푖푎푙 (푥)| for the real and complex solutions x = 2.54091, -0.16792 +/- 0.41579i.

Since equations [10] reduces m and b by 1, starting with any Jacobi polynomial there will be an eventual reduction to m = 0 an b=1. From the above analysis we can state:

Theorem 1: For any Jacobi Polynomial P[m,a,b,3] there exists a reduced polynomial of equal degree with all terms expressed as P[n,c,1,3] where n ranges from 0 to m and c increases from a to a + m.

Theorem 2: For any Jacobi Polynomial P[m,a,b,x] there is a reduced polynomial of equal degree (all terms expressed as P[n,c,1,x] that is equal to the Jacobi polynomial at x=3 and the difference is factored into terms (x-3) and a polynomial of degree m-1. Scalar Products of the Jacobi Polynomial

We now turn to the purpose of converting the Jacobi polynomial to reduced form. In equations [1] to [3] above the Jacobi polynomial can be expressed as a scalar product of two vectors, one vector from a row of the Pascal triangle and the other vector obtained either from the cycle index of a symmetry group or with a diagonal of the Chebyshev U polynomial. These products are found only if either or parameters a and b are zero. It is also possible to express the reduced Jacobi polynomial P[n, c, 1, 3] as a scalar product of a row from the Pascal triangle and a diagonal from the Chebyshev T polynomial. This polynomial was introduced in Part 1.

Definition 1: The reduced polynomial P[n, c, 1, 3] when expressed as P[n, n+c-n, 1, 3] has a value expressed by the (n-1)th row of the Pascal triangle and the n+c +1 diagonal of the Chebyshev T polynomial.

Here {1,1}, {1,2,1}, {1,3,3,1}, {1,4,6,4,1} … …. are rows 0, 1, 2, 3, 4…… and as an example P[4, 3, 1, 3]= P[4, 7-4, 1, 3] is expressed as the dot product of the 3rd row and the Pascal triangle and the 8th diagonal of the Chebyshev T polynomial. {1,4,6,4,1}. {1,15,98,364,840} = 2945. In Part 1 equations [43] and [44] express the monomial terms of the Chebyshev T polynomial. It can be shown that the jth term (where 1 is the 0th term) of the t = n+c+1 diagonal is

[16] (Binomial[t, 푗] + Binomial[t − 1, 푗]) ∗ 2^(푗 − 1)

In the reduced form the parameter n+c = a for all terms. The tth Chebyshev T diagonal then applies to all terms and only reduced in length by the value of n. As an example, the reduced form in equation [13] is expressed by the scalar products;

[17] P[4,3,3,3] = {1,4,6,4,1}. {1,15,98,364,840} + 2 ∗ {1,3,3,1}. {1,15,98,364} + 3 ∗ {1,2,1}. {1,15,98} + 4 ∗ {1,1}. {1,15} + 5 = 4809

For any fixed values of m and a, the reduced form requires m rows of the Pascal triangle, 1 diagonal of the Chebyshev T polynomial and m integer constants. The values of the integer constants only change values as the parameter b is increased or decreased. Here again we find a pattern associated with the Pascal triangle:

Definition 2- The ith coefficient of the reduced scalar product form is calculated from the 퐵푖푛표푚푖푎푙[푛 − 2 + 푖, 푛 − 2] with i = 0, 1, 2, 3, 4 ,…..(b-m+1).

Example: Expand the Jacobi Polynomial P[7,5,n,3] in reduced scalar product form.

[18] {1,7,21,35,35,21,7,1}. {1,25,288,2024,9680,33264,84480,160512} + Binomial[푛 − 1, 푛 − 2] ∗ {1,6,15,20,15,6,1}. {1,25,288,2024,9680,33264,84480} + Binomial[푛, 푛 − 2] ∗ {1,5,10,10,5,1}. {1,25,288,2024,9680,33264} + Binomial[푛 + 1, 푛 − 2] ∗ {1,4,6,4,1}. {1,25,288,2024,9680} + Binomial[푛 + 2, 푛 − 2] ∗ {1,3,3,1}. {1,25,288,2024} + Binomial[푛 + 3, 푛 − 2] ∗ {1,2,1}. {1,25,288} + Binomial[푛 + 4, 푛 − 2] ∗ {1,1}. {1,25} + Binomial[푛 + 5, 푛 − 2] This calculation demonstrates that the scalar product of a Jacobi polynomial can be formed without knowing the reduced form!

Reduction to a Scalar Product with the cycle index of a Symmetry group

We next show how to express the reduced form [4] using the shifted cycle index of a symmetry group. Using identity [10b] with b = 1;

푖=푚 P[푚, 푎, 1,3] = P[푚, 푎, 0,3] + ∑ P[푚 − 푖, 푎 + 푖, 0,3] [19] 푖=1 From the form of the scalar product in [18] it is possible to deduce the reduced form of each term. The pascal row is given by the second element in the vector which is m, and the second element of the Chebyshev T diagonal minus 1 is 2*(m+a). The first term in [18] is then P[7,5,1,3]. Using [19]

푖=7 P[7,5,1,3] = P[7,5,0,3] + ∑ P[7 − 푖, 5 + 푖, 0,3] [20] 푖=1 Unfortunately, this adds m new terms for each term of a reduced form. We can now use the shifted cycle index to calculate each term. As in the example shown in [3] above P[7,7-2,0,3] is calculated using th the 6 row of the Pascal triangle and the cycle index of S7 shifted by 5 units to the right. The corresponding terms in the summation of [20] are calculated by using the row number of the Pascal triangle for 7-i and the cycle index of the symmetry group S7-I shifted by a+i.

Theorem 3: For all Jacobi Polynomials P[m ,a, b, 3] there exists a reduced polynomial of equal degree with all terms expressible as P[ni,ci,0,3]. Each term can be represented by a scalar product of the ni th row of the Pascal triangle and the cycle index of a Sni shifted by ci units to the right. The first term is multiplied by unity and the subsequent terms are multiplied by the binomial of definition 2.

There exist m+1 terms in the reduced form of the vector product e.g. [18]. Transforming to the cycle index form there are Binomial[m+2,2] terms. Fortunately, once the first highest term is calculated the remaining terms are obtained by successively removing the highest m term. An example, the complete calculation of [18] in terms of the cycle index scalar product for P[7,5,n,3] is shown:

[21] {1,7,21,35,35,21,7,1}. {792,1716,3432,6435,11440,19448,31824,50388} + {1,6,15,20,15,6,1}. {924,1716,3003,5005,8008,12376,18564} + {1,5,10,10,5,1}. {792,1287,2002,3003,4368,6188} + {1,4,6,4,1}. {495,715,1001,1365,1820} + {1,3,3,1}. {220,286,364,455} + {1,2,1}. {66,78,91} + {1,1}. {12,13} + 1 + Binomial[푛 − 1, 푛 − 2] ∗ ({1,6,15,20,15,6,1}. {924,1716,3003,5005,8008,12376,18564} + {1,5,10,10,5,1}. {792,1287,2002,3003,4368,6188} + {1,4,6,4,1}. {495,715,1001,1365,1820} + {1,3,3,1}. {220,286,364,455} + {1,2,1}. {66,78,91} + {1,1}. {12,13} + 1) + Binomial[푛, 푛 − 2] ∗ ({1,5,10,10,5,1}. {792,1287,2002,3003,4368,6188} + {1,4,6,4,1}. {495,715,1001,1365,1820} + {1,3,3,1}. {220,286,364,455} + {1,2,1}. {66,78,91} + {1,1}. {12,13} + 1) + Binomial[푛 + 1, 푛 − 2] ∗ ({1,4,6,4,1}. {495,715,1001,1365,1820} + {1,3,3,1}. {220,286,364,455} + {1,2,1}. {66,78,91} + {1,1}. {12,13} + 1) + Binomial[푛 + 2, 푛 − 2] ∗ ({1,3,3,1}. {220,286,364,455} + {1,2,1}. {66,78,91} + {1,1}. {12,13} + 1) + Binomial[푛 + 3, 푛 − 2] ∗ ({1,2,1}. {66,78,91} + {1,1}. {12,13} + 1) + Binomial[푛 + 4, 푛 − 2] ∗ ({1,1}. {12,13} + 1) + Binomial[푛 + 5, 푛 − 2] The above analysis suggests a combinatoric role of the general Jacobi Polynomial evaluated at x = 3. We saw in Part 2 the connection of Delannoy numbers to the Jacobi polynomial P[m,a,0,3]. Based on the reduction of any Jacobi to a series of associated Delannoy numbers such as [21] a convoluted union of paths are likely represented. These paths suggest a connection to Brownian like motion in two dimensions.

a. Another polynomial sequence of the Jacobi polynomial P[m, a, 0, 1] can be obtained from the scalar product of the m- 1 and a-1 rows of the Pascal triangle. These rows are equivalent to the diagonal of the Chebyshev S polynomial. P[6,4,0,1] = 210 = {1,6,15,20,15,6,1}. {1,4,6,4,1,0,0}.