Change of Basis between Classical Orthogonal Polynomials D.A. Wolfram College of Engineering & Computer Science The Australian National University, Canberra, ACT 0200 [email protected] Abstract Classical orthogonal polynomials have widespread applications includ- ing in numerical integration, solving differential equations, and interpola- tion. Changing basis between classical orthogonal polynomials can affect the convergence, accuracy, and stability of solutions. We provide a general method for changing basis between any pair of classical orthogonal polynomials by using algebraic expressions called coefficient functions that evaluate to connection coefficients. The method builds directly on previous work on the change of basis groupoid. The scope has fifteen kinds of classical orthogonal polynomials including the classes of Jacobi, Gegenbauer and generalized Laguerre polynomials. The method involves the mappings to and from the monomials for these polynomial bases. Sixteen coefficient functions appear to be new for polynomials that do not have definite parity. We derive the remainder from seven sources in the literature. We give a complete summary of thirty coefficient functions. This enables change of basis to be defined algebraically and uniformly between any pair of classical orthogonal polynomial bases by using a vector dot product to compose two coefficient functions to give a third. A definition of Jacobi polynomials uses a basis of shifted monomials. We find a key new mapping between the monomials and Jacobi polynomi- arXiv:2108.13631v1 [math.CA] 31 Aug 2021 als by using a general mapping between shifted monomials and monomials. It yields new mappings for the Chebyshev polynomials of the third and fourth kinds and their shifted versions. 1 Introduction and Related Work Change of basis of a finite vector space has numerous significant applications in scientific computing and engineering. Applications of change of basis with clas- sical orthogonal polynomials include improving properties of spectral methods MSC: Primary 33C45; Secondary 15A03, 20N02 1 for solving differential equations numerically. These properties include better convergence [6, 9], lower computational complexity [11, 15], and better numeri- cal stability [15]. Previous related work gives recurrence relations for finding connection coef- ficients for orthogonal polynomials, e.g., [14], for Jacobi polynomials [13], or by other methods [5, 7] and for special cases [4, 6]. This work derives coefficient functions that evaluate to connection coeffi- cients for mappings between the classical orthogonal polynomials and the mono- mials. We write the coefficient functions as algebraic expressions. From the framework of the change of basis groupoid [20], a vector dot product equa- tion enables us to compose coefficient functions in order to derive a coefficient function for any pair of classical orthogonal polynomials. 2 Classifications of Orthogonal Polynomials Thirteen kinds of orthogonal polynomials are classified by Koornwinder et al. [10, §18.3] as classical orthogonal polynomials: (α,β) • Jacobi, Pn (x) (λ) • Gegenbauer, Cn (x) • Chebyshev of the first to fourth kinds: Tn(x),Un(x), Vn(x), Wn(x) ∗ ∗ • Shifted Chebyshev of the first and second kinds: Tn (x),Un(x) ∗ • Legendre and Shifted Legendre, Pn(x), Pn (x) (α) • Generalised Laguerre, Ln (x) • Physicists’ and Probabilists’ Hermite: Hn(x), and Hen(x). The Jacobi, Gegenbauer and generalized Laguerre polynomials depend on the respective parameters, α and β, λ, and α. Andrews and Askey [3] give a more general classification of the classical orthogonal polynomials and stated “there are a number of different places to put the boundary for the classical polynomials”. We also include the shifted Chebyshev polynomials of the third and fourth ∗ ∗ kinds here, Vn (x) and Wn (x), because the classification is not fixed. 3 Related Classical Orthogonal Polynomials We shall use some of the properties described here that relate different kinds of classical orthogonal polynomials. Specifically, Chebyshev polynomials of the third and fourth kinds and Jacobi polynomials, and Chebyshev polynomials of the second kind and Legendre polynomials as special cases of Gegenbauer polynomials. 2 The Jacobi and Gegenbauer polynomials are related to some of the other classical orthogonal polynomials. The Gegenbauer polynomials are a special 1 1 case of the Jacobi polynomials where α = β, α = λ 2 and α> 1 or λ> 2 . From Szeg¨o[16, §4.7], − − − Γ(α + 1) Γ(Γ(n +2α + 1) C(λ)(x)= P (α,α)(x) (1) n 2α +1 n + α +1 n 1 λ 1 1 Γ( + 2 ) Γ(n +2λ) (λ− 2 ,λ− 2 ) = 1 Pn (x) (2) Γ(2λ) Γ(n + λ + 2 ) Chebyshev polynomials of the first kind are related to the Gegenbauer poly- nomials, e.g., Arfken [2]: (α) T0(x)=C0 (x) (3) (α) n Cn (x) Tn(x)= lim where n> 0. (4) 2 α→0 α The Chebyshev polynomials of the second kind are a special case of the (1) Gegenbauer polynomials with Un(x)= Cn (x). Chebyshev polynomials of the third and fourth kinds, Vn(x) and Wn(x), are special cases of the Jacobi poly- nomials, e.g., [12, §1.2.3]. The Legendre polynomials are also a special case of Gegenbauer polynomials 1 ( 2 ) with Pn(x)= Cn (x). 4 Definitions and Background Equations We consider a method of defining algebraically a change of basis mapping be- tween finite bases of classical orthogonal polynomials or the monomials. The method uses the matrix dot product of change of basis matrices. From the change of basis groupoid [20], given two change of basis matrices MtsMsv, we have Mtv = MtsMsv. (5) The elements of Mtv are formed from the vector dot products of the rows of Mts and the columns of Msv. All classical orthogonal polynomials and the monomials satisfy the following condition. It leads to an optimization of the vector dot products. Definition 1. Let V be a vector space of finite dimension n > 0 and B1 and B2 be bases of V . The basis B2 is a triangular basis with respect to B1 if and only if there is a permutation v1,...,vn of the coordinate vectors of B2 with respect to B1 such that the n n{ matrix v}T ,...,vT is an upper triangular matrix. × 1 n 3 Definition 2. Suppose that s and t are triangular bases of a vector space V . The mapping : s t satisfies T → n sn = α(n, k)tn−k (6) k X=0 where α(n, k) R, i.e., each basis vector of s is a unique linear combination of the basis vectors∈ of t. The function α is called a coefficient function and it evaluates to a connection coefficient. If α(n, k) is the coefficient function in equation (6), then the elements of the change of basis matrix Mts are defined by α(j, j i) if j i m = − ≥ (7) ij 0 if j<i. where 0 i, j n. ≤ ≤ Suppose that s, t and v are triangular bases, α1(n, k) is the coefficient func- tion for a change of basis matrix Mts and α2(n, k) is a coefficient function for Msv following equation (7). Then k α (n, k)= α (n v, k v)α (n, v) (8) 3 1 − − 2 v=0 X where 0 k n, is the coefficient function for Mtv. The basis≤ ≤s is called the exchange basis [20]. We shall use the monomials as the exchange basis. 4.1 Parity and Bases Six of the classical orthogonal polynomials have definite parity, and nine do not: the Jacobi polynomials, the generalized Laguerre polynomials, Chebyshev polynomials of the third and fourth kinds, and shifted orthogonal polynomials: T ∗, U ∗, V ∗, W ∗ and P ∗. (α,β) For example, the Jacobi Polynomials Pn (x) can be expressed in terms x−1 x−1 2 x−1 n of the basis 1, ( 2 ), ( 2 ) ,..., ( 2 ) , e.g., [10, equation 18.5.7]. The gen- { (α) } eralised Laguerre Polynomials Ln (x) can be expressed in terms of the basis 1, x, x2,...,xn . { The Chebyshev} Polynomials of the second kind have definite parity and they can be defined by n ⌊ 2 ⌋ n k −2k n k n−2k Un(x)=2 ( 1) 2 − x . − k k X=0 where n 0. When n is even, Un(x) can be expressed in terms of the ba- 2 ≥ 4 n sis 1, x , x ,...,x . When n is odd, Un(x) can be expressed in terms of x, x{3, x5,...,xn . } { } 4 Equation (8) can be optimized when the bases t and v have definite parity and we are concerned with finding the coefficient function for Mtv with basis vectors that all have either even parity or odd parity. In the example above, we have a coefficient function β(n, k) for non-zero coefficients: n k β(n, k) = ( 1)k2n−2k − − k where 0 k n . ≤ ≤⌊ 2 ⌋ Given β1(n, k) and β2(n, k) corresponding to the change of basis matrices Mts and Msv respectively, we have k β (n, k)= β (n 2v, k v)β (n, v) (9) 3 1 − − 2 v=0 X where 0 k n . This optimization excludes terms that are zero [20, §5.1]. ≤ ≤⌊ 2 ⌋ From equation (5), Mtv = MtsMsv, parity is not relevant if at least one of t or v does not have definite parity. In this case, the following equation can be used when β is known, but α is not. k β(n, 2 ) if k is even α(n, k)= (10) 0 if k is odd. The upper triangle of the associated change of basis matrix has zero and non-zero alternating elements. 5 Classical Orthogonal Polynomials and the Monomials We summarize here results of mappings between the classical orthogonal poly- nomials and the monomials. With the monomials as the exchange basis, we can find mappings between any pair of classical orthogonal polynomials.