arXiv:2108.13631v1 [math.CA] 31 Aug 2021 ia rhgnlplnmasicueipoigpoete fspectr of properties improving basis include of ap change polynomials significant of orthogonal numerous Applications sical has engineering. space vector and finite computing a scientific Work of basis Related of Change and Introduction 1 S:Piay3C5 eodr 50,20N02 15A03, Secondary 33C45; Primary MSC: tyed e apnsfrteCeyhvplnmaso h t the of versions. polynomials shifted Chebyshev their the and for kinds fourth mappings and new monomials poly yields shifted Jacobi between It mapping and general monomials a the using between by mapping als new key a find We o rdc ocmoetocecetfntost ieathir a give to functions coefficient two compose to b bases product summ polynomial dot orthogonal complete classical of a pair any between give We literature. the functions. th coefficient derive in thirty We sources parity. definite appea seven have functions not from do coefficient that Sixteen polynomials for bases. polynomial these pol Laguerre group generalized polynomials basis and orthogonal Gegenbauer of classical Jacobi, change of of the kinds classes on fifteen work has previous scope on coefficient directly exp connection builds to algebraic evaluate using that by functions coefficient polynomials orthogonal classical of solutions. of stability polynom and accuracy, orthogonal convergence, classical the between basis equatio Changing differential solving tion. integration, numerical in ing lsia rhgnlPolynomials Orthogonal Classical ento fJcb oyoil ssabsso hfe mon shifted of basis a uses polynomials Jacobi of definition A fo monomials the from and to mappings the involves method The hsealscag fbsst edfie lerial n u and algebraically defined be to basis of change enables This pa any between basis changing for method general a provide We applica widespread have polynomials orthogonal Classical h utainNtoa nvriy abra C 0200 ACT Canberra, University, National Australian The hneo ai between Basis of Change olg fEgneig&Cmue Science Computer & Engineering of College [email protected] ..Wolfram D.A. Abstract 1 s n interpola- and ns, sn vector a using y .Temethod The s. esoscalled ressions ynomials. ascnaffect can ials nldn the including in includ- tions ob new be to r remainder e monomials. d. i.The oid. idand hird niformly omials. lctosin plications r of ary lmethods al nomi- ihclas- with ir r 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

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. In the first table below, the domain is in the first row of a table and the range is the monomials. In the second table, the range is in the first row and the domain is the monomials. Where a citation is given, a related formula has been given in the literature. The boldface equation numbers in the tables refer to equations derived below.

5 (α,β) (λ) ∗ ∗ ∗ ∗ ∗ (α) P C T U V W T U V W P P L H He M (16) [1] [1, 12] [1] [8] & (23) [8] & (25) (32) (33) (34) (35) [1] (36) [1] [1] [1]

Table 1: Mappings to the Monomials

(α,β) (λ) ∗ ∗ ∗ ∗ ∗ (α) P C T U V W T U V W P P L H He M (19) [1, 10] [12, 17] [10] (28) (30) (38) & (39) (41) (42) (43) [10, 19] (45) [10] [10, 18] [18]

Table 2: Mappings from the Monomials

We are concerned with defining algebraic expressions for the elements in change of basis matrices. From Tables 1 and 2 above, we can use equation (8) to find the coefficient function for any pair of bases that are classical orthogonal polynomials.

5.1 Shifted Monomials and Polynomials The shifted classical orthogonal polynomials and the Jacobi polynomials are sometimes defined, e.g., [10, Table 18.3.1] and [12, §1.3.1], by using a basis of shifted monomials. To express them in terms of the monomials involves changes of bases between shifted monomials and monomials, which we now discuss. The intervals of orthogonality of the shifted classical orthogonal polynomials are (0, 1) instead of ( 1, 1). They can be defined by applying the polynomial function for the unshifted− polynomials to 2x 1 instead of x. Similarly, the interval of orthogonality of the− Jacobi polynomials is ( 1, 1) [10, Table 18.3.1]. They can be viewed as shifted polynomials where the polyn− omial x−1 function has been applied to 2 instead of x, so that the interval before the application was ( 1, 0). Generally, applying− a classical orthogonal polynomial function to cx + d instead of x results in a shift of the interval of orthogonality from (a,b) to a−d b−d ( c , c ). To find the change of basis mappings between the shifted polynomials and the monomials, we first find the change of basis mappings between the mono- mials and the basis 1, cx + d, (cx + d)2,..., (cx + d)n . From the Binomial Theorem, the mapping{ to the monomials is given by the} coefficient function n α(n,k,c,d)= cn−kdk. (11) k   For example, when n = 3, c = 1 and d = 2 the change of basis matrix is 3 − 3 1 2 4 8 − 3 9 − 27 0 1 4 4  3 − 9 9     0 0 1 2   9 − 9      0 0 0 1   27   

6 and from the fourth column x 2 3 1 2 4 8 − = x3 x2 + x . 3 27 − 9 9 − 27   The inverse mapping from the monomials is given by the coefficient function 1 d n α(n, k, , )= c−n( d)k. (12) c − c k −   Similarly, when n = 3, c = 1 and d = 2 , the change of basis matrix is 3 − 3 12 4 8

0 3 12 36     0 0 9 54       0 0 0 27   and from the fourth column   x 2 3 x 2 2 x 2 x3 = 27 − + 54 − + 36 − +8. 3 3 3       Equations (11) and (12) can be used to give change of basis matrices for the mappings between the monomials and the four kinds of shifted Chebyshev poly- nomials, and the shifted Legendre polynomials. For example, with the shifted Chebyshev polynomials of the second kind, ∗ we have Un(x) = Un(2x 1), so that c = 2 and d = 1. The mapping from the basis of shifted Chebyshev− polynomials of the second− kind to the monomials follows from equations (11), Abramowitz and Stegun [1, equation 22.3.7] and equation (8). Specifically, from equation (11), n α (n, k)= 2n−k( 1)k. 1 k −   From equations 22.3.7 and (10),

k k n− 2 n−k 2 k 2 ( 1) if k is even 2 − α (n, k)= 2 
 0 if k is odd.

Hence from equation (8), when n = 3 the change of basis matrix is 1 2 3 4 − − 0 4 16 40 −     0 0 16 96  −      0 0 0 64    

7 and from the fourth column U (2x 1) = 64x3 96x2 + 40x 4. 3 − − − The inverse mapping from the monomials to the shifted Chebyshev or shifted Legendre polynomials uses equation (12), the mapping from monomials to the unshifted polynomials, and equation (8). For example, for the shifted Chebyshev polynomials of the first kind, we use the mapping from the monomials to the Chebyshev Polynomials of the first kind [12, 17] and equation (10) to give

n 1−n k 2 if k is even and 0 k

 n
−n α1(n, k)= k 2 if k is even and k = n  2  
0 if k is odd.   With c = 2 and d = 1, equation (12) gives − n α (n, k)=2−n . 2 k   The change of basis matrix is 1 3 5 1 2 8 16

 1 1 15  0 2 2 32     0 0 1 3   8 16      0 0 0 1   32    whose elements are j−i α (j, j i)= α (j v, j i v)α (j, v) 3 − 1 − − − 2 v=0 X where 0 i j 3 and 0 otherwise. From the third and fourth columns, we have ≤ ≤ ≤ 1 1 3 x2 = T (2x 1)+ T (2x 1)+ 8 2 − 2 1 − 8 1 3 15 5 x3 = T (2x 1) + T (2x 1)+ T (2x 1)+ . 32 3 − 16 2 − 32 1 − 16 5.2 Jacobi Polynomials A definition of the Jacobi polynomials [10, equation 18.5.7] is

n l (n + α + β + 1)l(α + l + 1)n l z 1 P (α,β)(z)= − − n l!(n l)! 2 l X=0 −  

8 which uses the Pochhammer symbol or rising factorial. This can be defined as z−1 z−1 2 a polynomial in the basis 1, 2 , 2 ,... . This basis is a triangular basis following Definition (1). We{ can change basis} to the monomials.
z−1 z−1 2 From equation (11), the change of basis matrices from 1, 2 , 2 ,... to monomials has coefficient function { }
n α(n, k)= 2−n( 1)k (13) k −   1 1 where c = 2 and d = 2 . For example, when−n = 4, the change of basis matrix is

1 1 1 1 − 2 4 8 0 1 1 3  2 − 2 8     0 0 1 3   4 − 8      0 0 0 1   8    (α,β) The coordinates of P3 (z) in the domain basis written as a vector are

1 6 (1 + α)(2 + α)(3 + α)

 1  2 (2 + α)(3 + α)(4 + α + β)      1 (3 + α)(4 + α + β)(5 + α + β)   2       1 (4 + α + β)(5 + α + β)(6 + α + β)  6    The product of the change of basis matrix above with this vector, gives the coordinates of the Jacobi polynomial in the basis of the monomials expressed as a vector. For example, the expression below is the third element, and it is the coeffi- 2 (α,β) cient of the term in z of P3 (z): 1 1 (3 + α)(4 + α + β)(5 + α + β) (4 + α + β)(5 + α + β)(6 + α + β) 8 − 16 which equals

5a 9α2 α3 5β α2β 9β2 αβ2 β3 + + + . 4 16 16 − 4 16 − 16 − 16 − 16 It follows that

n n−m n−m ( 1) (1 + α + β + n)n l( α n)l P (α,β)(z)= − − − − zm (14) n 2n−l (n m l)! m! l! m=0 l ! X X=0 − −

9 or equivalently

Γ(α + n + 1) P (α,β)(z)= n n!Γ(α + β + n + 1) n n−m n l n Γ(α + β +2n l + 1) ( 2)l−n( 1)m − − zm − − m n l Γ(α + n l + 1) m=0 l=0   −  − ! X X (15)

Therefore, we have,

( 1)n−kΓ(α + n + 1) α(n, k)= − n!Γ(α + β + n + 1) k n l n Γ(α + β +2n l + 1) ( 2)l−n − − . (16) − n k n l Γ(α + n l + 1) l X=0  −  −  − For the inverse mapping, equation 18.18.15 of Koornwinder et al. [10] is

n n 1+ x (β + 1)n α + β +2l +1 (α + β + 1)l(n l + 1)l (α,β) = − Pl (x). 2 (α + β + 2)n α + β +1 (β + 1)l(n + α + β + 2)l l   X=0 1+x 1+x 2 This defines a change of basis from 1, 2 , 2 ,... to the Jacobi poly- nomials. From equation (12), the change{ of basis matrices} from the monomials 2
to 1, 1+x , 1+x ,... has coefficient function { 2 2 }
n α(n, k)= 2n−k( 1)k (17) k −   1 1 where c = 2 and d = 2 . This gives

n n m α + β +2m +1 − n (β + 1) zn = 2n−l ( 1)l n−l α + β +1 n l − (α + β + 2)n l m=0 l − X X=0  −  (α + β + 1)m(n l m + 1)m (α,β) − − Pm (z). (18) (β + 1)m(n l + α + β + 2)m −  We have

α(n, k) = (α + β + 2(n k) + 1)(α + β + 2)n k − − −1 k n (β +1+ n k)k l (k l + 1)n k 2n−l ( 1)l − − − − . n l − (α + β + 2)n−l (n l + α + β + 2)n−k l=0  −  − X (19)

10 5.3 Chebyshev Polynomials of the Third and Fourth Kinds These classical orthogonal polynomials can be respectively defined from the Jacobi polynomials [12]

2n 1 1 2 (− 2 , 2 ) Vn(x)= 2n Pn (x) (20) n 2n 1 , 1 2
( 2 − 2 ) Wn(x)= 2n Pn (x). (21) n

Vn(x) and Wn(x) have terms in all powers
of x up to n when expressed in terms of monomials. From equation (14), we have

n n n−m 1 2 n−m l (1 + n)n−l( 2 n)l m Vn(x)= ( 1) 2 − x (22) 2n − (n l m)! m! l! n m=0 l ! X X=0 − − so that

n k k 1 2 ( 1) (1 + n)n−l( n)l α(n, k)= − 2l 2 − . (23) 2n (n k)! (k l)! l! n l − X=0 − Similarly,

n n n−m 1 2 n−m l (1 + n)n−l( 2 n)l m Wn(x)= ( 1) 2 − − x (24) 2n − (n l m)! m! l! n m=0 l ! X X=0 − − so that

n k k 1 2 ( 1) (1 + n)n−l( n)l α(n, k)= − 2l − 2 − . (25) 2n (n k)! (k l)! l! n l − X=0 − Explicit formulas for these
polynomials were also given by another approach by Dewi, Utama and Animah [8]. They are

n k 2k−n−1 n−k 2k−n−1 2k n Vn(x)= 2 ( 1) x 2x − n k − − k k=⌈ n ⌉ X2  −    n k 2k−n−1 n−k 2k−n−1 2k n Wn(x)= 2 ( 1) x 2x + − n k − k k=⌈ n ⌉ X2  −    where n 2. These formulas also apply when n = 1. In the≥ converse direction, from equation (19), let

n−m 3 n−l n l ( 2 )n−l (1)m(n l m + 1)m f(m,n)= 2 ( 1) 3 − − n l − (2)n l ( ) (n l + 2) l − 2 m m X=0  −  −

11 so that from equation (20), we obtain

n 2m xn = (2m + 1) m f(m,n)V (x). (26) 22m m m=0
X This can be simplified by using Γ(x + n) 1 (2n 1)!! (x) = and Γ( + n)= − √π n Γ(x) 2 2n to give

n n−m ( 1)l (2(n l) + 1)!! n! xn = − − V (x). (27) (n l m)! (n l + m + 1)! l! m m=0 l ! X X=0 − − − The coefficient function for the change of basis from monomials to Chebyshev polynomials of the third kind is therefore

k ( 1)l (2(n l) + 1)!! α(n, k)= n! − − . (28) (k l)! (2n l k + 1)! l! l X=0 − − − Similarly, we obtain

n n−m ( 1)l (2(n l) 1)!! n! xn = (2m + 1) − − − W (x). (29) (n l m)! (n l + m + 1)! l! m m=0 l ! X X=0 − − − The coefficient function for the change of basis matrix from Wn(x) to the monomials is k ( 1)l (2(n l) 1)!! α(n, k) = (2(n k)+1)n! − − − . (30) − (k l)! (2n l k + 1)! l! l X=0 − − − For example, when n = 4, we have

1 1 1 3 3 − 2 2 − 8 8 0 1 1 3 1  2 − 4 8 − 4     0 0 1 1 1   4 − 8 4      0 0 0 1 1   8 − 16      00 0 0 1   16    and from the fifth column, for example, 1 1 1 1 3 x4 = W (x) W (x)+ W (x) W (x)+ W (x). 16 4 − 16 3 4 2 − 4 1 8 0

12 6 Shifted Classical Orthogonal Polynomials

The shifted classical orthogonal polynomials include the shifted Chebyshev poly- ∗ ∗ nomials of the first kind, Tn (x), the fourth kind, Wn (x), and the shifted Legen- ∗ dre polynomials, Pn (x). All of them can be defined by applying the unshifted ∗ polynomial function to the argument 2x 1 instead of x, e.g., Tn (x)= Tn(2x 1). They each have (0, 1) as their domain− of orthogonality [10, Table 18.3.1].− We note that the method here is general and can be used for any linear shift of the form cx + d and for any triangular polynomial basis.

6.1 Mappings to the Monomials With these definitions, and the definitions of the unshifted polynomials in terms of the monomials, the shifted polynomials can be expressed using the basis 1, 2x 1, (2x 1)2,..., (2x 1)n . To express the shifted polynomials in terms of{ the− monomials,− we use equation− } (11) with c = 2 and d = 1, i.e. − n α(n, k)= ( 1)k2n−k (31) k −   and then apply equation (9) because these polynomials have definite parity. For shifted Chebyshev polynomials of the first kind we have

n ⌊ 2 ⌋ n n k −2k (n k 1)! n−2k Tn(x)= 2 ( 1) 2 − − x 2 − k!(n 2k)! k X=0 − so that n (n k 1)! β (n, k)= 2n( 1)k2−2k − − . 2 2 − k!(n 2k)! − Therefore,

⌊ k ⌋ 2 n 2v n (n v 1)! α(n, k)= − ( 1)k−2v2n−k 2n( 1)v2−2v − − k 2v − 2 − v!(n 2v)! v=0 X  −  − ⌊ k ⌋ n 2 n 2v (n v 1)! = − ( 1)k−v22(n−v)−k − − (32) 2 k 2v − v!(n 2v)! v=0 X  −  − and α(0, 0) = 1. We have

n ∗ n−k Tn (x)= α(n, k)x . k X=0

13 Similarly, for shifted Chebyshev polynomials of the second kind, we obtain

⌊ k ⌋ 2 n 2v n v α(n, k)= − ( 1)k−2v2n−k2n−2v − ( 1)v k 2v − v − v=0 X  −    ⌊ k ⌋ 2 n 2v n v = − − ( 1)k−v22(n−v)−k. (33) k 2v v − v=0 X  −   The shifted Chebyshev polynomials of the third and fourth kinds, V ∗ and ∗ ∗ ∗ W , can be similarly defined by Vn (x)= Vn(2x 1) and Wn (x)= Wn(2x 1), e.g., [12, equation (1.27)] − − ∗ For the mapping from the monomials to Vn (x), we apply equation (8) with α1 from equation (31) and α2 from equation (23) to give :

2n−k k v 1 k 2 l (1 + n)n−l( 2 n)l α3(n, k) = ( 1) 2 − . (34) − 2n n k (k v)! (v l)! l! n ( )! v=0 l − X X=0 − −
∗ For the similar mapping to Wn (x) we use α2 from equation (25) instead to obtain:

2n−k k v 1 k 2 l (1 + n)n−l( 2 n)l α3(n, k) = ( 1) 2 − − . (35) − 2n n k (k v)! (v l)! l! n ( )! v=0 l − X X=0 − − For shifted Legendre polynomials,
we have

⌊ k ⌋ 2 n 2v 2n 2v n α(n, k)= − ( 1)k−2v2n−k2−n − ( 1)v k 2v − n v − v=0 X  −     ⌊ k ⌋ 2 n 2v 2n 2v n =2−k − − ( 1)k−v. (36) k 2v n v − v=0 X  −    6.2 Mappings from the Monomials In the converse direction, we apply the change of bases from

1, 2x 1, (2x 1)2,..., (2x 1)n { − − − } ∗ ∗ ∗ to the shifted polynomials Tn (x), Un(x) and Pn (x). We first consider the change of basis from the monomials to 1, 2x 1, (2x 1)2,..., (2x 1)n . This mapping follows from equation{ − (12) with− c = 2 and −d = }1 to give − n α (n, k)= 2−n. (37) 2 k  

14 The change of basis from monomials to Chebyshev polynomials of the first kind is given by the coefficient function

n 21−n if 0 2k

k α (n, k)= α (n v, k v)α (n, v) 3 1 − − 2 v=0 k−vXeven k k v n = β (n v, − ) 2−n. 1 − 2 v v=0   k−vXeven This gives

k n v n α (n, k)=21−2n − 2v if 0 k

k n v n α (n, k)=2−2n − 2v if k = n (39) 3 k−v v v=0  2   k−vXeven and n n ∗ x = α3(n, k)Tn−k(x). k X=0 For the Chebyshev polynomials of the second kind, the mapping from the monomials has the coefficient function

−n n! β1(n, k)=2 (n 2k + 1) . (40) − (2)n−kk!

This follows from Koornwinder et al. [10, equation 18.18.17] with λ =1. Ina

15 ∗ similar way to the case for Tn (x) above, we have

k α (n, k)= α (n v, k v)α (n, v) 3 1 − − 2 v=0 k−vXeven k k v n = β (n v, − ) 2−n 1 − 2 v v=0   k−vXeven k v −2n 2 =2 (1 + n k)n! k−v . (41) − (2) v+k v!( )! v=0 n− 2 2 k−vXeven

∗ For the mapping from Vn (x) from the monomials, we use α1 from equa- tion (28) and α2 from equation (37) to give:

k k−v ( 1)l (2(n v l) + 1)!! α (n, k)=2−nn! − − − . (42) 3 (k v l)! (2n v l k + 1)! v! l! v=0 l X X=0 − − − − − ∗ Similarly for Wn (x) we use α1 from equation (30):

k k−v ( 1)l (2(n v l) 1)!! α (n, k)=2−n(2(n k)+1)n! − − − − . 3 − (k v l)! (2n v l k + 1)! v! l! v=0 l=0 − − − − − X X (43) For the shifted Legendre polynomials, we have from Koornwinder et al. [10, 1 equation 18.18.17] with λ = 2 that

−n n! β1(n, k)=2 (1+2n 4k) 3 (44) − ( 2 )n−kk! so that

k α (n, k)= α (n v, k v)α (n, v) 3 1 − − 2 v=0 k−vXeven k k v n = β (n v, − ) 2−n 1 − 2 v v=0   k−vXeven k v −2n 2 =2 (2(n k)+1)n! 3 k−v . (45) − ( ) v+k v!( )! v=0 2 n− 2 2 k−vXeven For example,

4 ∗ ∗ ∗ ∗ ∗ x =α3(4, 0)P4 (x)+ α3(4, 1)P3 (x)+ α3(4, 2)P2 (x)+ α3(4, 3)P1 (x)+ α3(4, 4)P0 (x) 1 1 2 2 1 = P (2x 1) + P (2x 1)+ P (2x 1)+ P (2x 1)+ . 70 4 − 10 3 − 7 2 − 5 1 − 5

16 7 Examples

We give two examples of derivations of coefficient functions for change of basis between Jacobi polynomials, and Physicist’s Hermite polynomials to the shifted Legendre polynomials. The coefficient function for Jacobi polynomials has four extra parameters which are the parameters of the Jacobi polynomials: two for the those in the domain basis and two for the range basis. By swapping these parameters, we find the inverse of the change of basis matrix. The example of deriving the coefficient function for the change of basis from Physicist’s Hermite polynomials to the shifted Legendre polynomials uses equa- tion (8) because the shifted Legendre polynomials do not have definite parity.

7.1 Jacobi Polynomials From equation (14), we have

k k ( 1) (1 + α + β + n)n l( α n)l α (n,k,α,β)= − − − − . (46) 2 2n−l (k l)! (n k)! l! l X=0 − − From equation (19), let k = n m, so that − k γ + δ + 2(n k)+1 n−l n l (δ + 1)n−l α1(n,k,γ,δ)= − 2 ( 1) γ + δ +1 n l − (γ + δ + 2)n l l − X=0  −  (γ + δ + 1)n k(k l + 1)n k − − − . (47) (δ + 1)n k(n l + γ + δ + 2)n k − − −  Hence from equation (8), it follows that

k α (n,k,α,β,γ,δ)= α (n v, k v,γ,δ)α (n,v,α,β). (48) 3 1 − − 2 v=0 X where 0 k n. This≤ equation≤ gives the elements of the upper triangular change of ba- (α,β) sis matrix for the mapping from the basis Pn (z),n 0 to the basis (γ,δ) { ≥ } Pn (z),n 0 . { Another coefficient≥ } function for Jacobi Polynomials can be derived from equation 18.18.14 of Koornwinder et al. [10]. These coefficient functions gener- alize three solutions for special cases by Askey [4] who used another method: equation (7.32) where α = γ, equation (7.33) where β = δ, and equation (7.34) where α = β and γ = δ. The inverse change of basis matrix has the property that it can be found by interchanging α and γ with β and δ, respectively, by using α3(n,k,γ,δ,α,β). For example, let n = 4, α = 1 and β = 8, γ = 2, δ = 7. The change of basis (2,7) (1,8) matrix from Pn (z), 0 n 4 to the basis Pn (z), 0 n 4 is { ≤ ≤ } { ≤ ≤ }

17 9 15 15 1 1 11 22 26

 12 10 10  0 1 11 11 13     0 0 1 7 77   6 78      00 0 1 16   13      000 0 1      and from the fourth column, 7 10 15 P (2,7)(z)= P (1,8)(z)+ P (1,8)(z)+ P (1,8)(z)+ P (1,8)(z). 3 3 6 2 11 1 22 0 7.2 Physicist’s Hermite Polynomials to Shifted Legendre Polynomials The coefficient function in this example can be found from equation (8) because shifted Legendre polynomials do not have definite parity. The coefficient function for the change of basis from the monomials to shifted Legendre polynomials is from equation (45) for α1(n, k). From Table 1 the coefficient function for the change of basis from Physi- cist’s Hermite polynomials to the monomials is derived from Abramowitz and Stegun [1, equation 22.3.10]. This is n! β (n, k) = ( 1)k2n−2k . (49) 2 − k!(n 2k)! − From equation (10),

v 2 n−v n! ( 1) 2 ( v )!(n−v)! if v is even − 2 α (n, v)= (50) 2   0 if v is odd. From equation (8), we have

k α (n, k)= α (n v, k v)α (n, v). 3 1 − − 2 v=0 X It follows that 5 ∗ H5(x)= α3(5, v)P5−v(x) v=0 X 8 8 32 640 4 76 = P ∗(x)+ P ∗(x) P ∗(x) P ∗(x) P ∗(x)+ P ∗(x) 63 5 7 4 − 9 3 − 21 2 − 7 1 3 0 =32x5 160x3 + 120x. −

18 8 Conclusion

Previous work [20] introduced a change of basis groupoid and provides the frame- work for this work. We reduced the scope here to fifteen classical orthogonal polynomials including the classes of Jacobi polynomials, Gegenbauer polynomi- als and generalized Laguerre polynomials. We used the monomials as the exchange basis and gave thirty algebraic expressions called coefficient functions that evaluate to connection coefficients for the mappings to and from the monomials. The appendix summarizes them. With this library of coefficient functions, the framework enables us to derive a coefficient function for the mapping between any pair of classical orthogonal polynomial bases. We do this using equation (8), or its optimizations [20] when at least one of the bases has definite parity. Of the thirty algebraic expressions, eighteen relate to classical orthogonal polynomials that do not have definite parity. Sixteen of these expressions seem new. We used the change of basis between shifted monomials and the monomi- als. This enabled us to map the basis of Jacobi polynomials to and from the monomials. This mapping led to a similar one for Chebyshev polynomials of the third and fourth kinds by using a definition of them based on the Jacobi polynomials. The technique also produced mappings for five shifted classical orthogonal polynomials: T ∗, U ∗, V ∗, W ∗ and P ∗. The technique allows for any linear shift cx + d, and it applies to any triangular polynomial basis over C[x]. We then gave examples for changes of bases between Jacobi polynomials, and from Physicist’s Hermite polynomials to the shifted Legendre polynomials.

Acknowledgment

I am grateful to the College of Engineering & Computer Science at The Aus- tralian National University for research support.

References

[1] Abramowitz, M., Stegun, I.A. (Eds.) (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 10th printing, National Bureau of Standards, Department of Commerce, USA. [2] Arfken, G. (1985). Mathematical Methods for Physicists, Third Edition, Or- lando, Florida: Academic Press. §13.3, pp 731–732. [3] Andrews, G.E., Askey, R. (1985). Classical orthogonal polynomials. In: Brezinski, C, Draux, A., Magnus, A.P., Maroni, P., Ronveaux, A. eds., Polynˆomes Orthogonaux et Applications: Proceedings of the Laguerre Sym- posium held at Bar-le-Duc, October 15–18, 1984, Lecture Notes in Math., vol. 1171, Berlin, Germany: Springer, pp. 36–62.

19 [4] Askey, R. (1975). Orthogonal Polynomials and Special Functions. Philadel- phia: SIAM. [5] Bella, T., Reis, J. (2015). The spectral connection matrix for any change of basis within the classical real orthogonal polynomials. Math. 3: 382–397. [6] Boyd, J.P., Petschek, R. (2014). The relationships between Chebyshev, Leg- endre and Jacobi polynomials: The generic superiority of Chebyshev poly- nomials and three important exceptions, J. Sci. Comput. 59: 1–27. [7] Chaggara, H., Mbarki, R. (2016). On connection sequence for equivalent polynomial sets. Integral Transforms Spec. Funct. 27(4): 1–16. [8] Dewi, I.P., Utama, S., Aminah, S. (2017). Deriving the explicit formula of Chebyshev polynomials of the third kind and fourth kind. Proceedings of the Third International Symposium on Current Progress in Mathematics and Sciences 2017 (ISCPMS 2017), AIP Conference Proceedings 2023, 020202 (2018). https://doi.org/10.1063/1.5064199, 7 pp. [9] Hale, N., Olver, S. (2018). A fast and spectrally convergent algorithm for rational-order fractional integral and differential equations. SIAM J. Sci. Comput. 40: A2456–A2491. [10] T.H. Koornwinder, R. Wong, R. Koekoek, R.F. Swarttouw. (2020). Chap- ter 18 Orthogonal Polynomials, Digital Library of Mathematical Functions, National Institute of Standards and Technology, U.S. Department of Com- merce. Available at: https://dlmf.nist.gov/18 [11] Fortunato, D., Hale, N., Townsend, A. (2021). The ultraspherical spectral element method. J. Comput. Phys. 436: 110087. [12] Mason, J.C., Handscomb, D.C. (2002). Chebyshev Polynomials. New York: Chapman and Hall/CRC. [13] Lewanowicz, S. (1992). Quick construction of recurrence relations for the Jacobi coefficients. J. Comput. Appl. Math. 43: 355-–372. [14] Maroni, P., da Roncha, Z. (2013). Connection coefficients for orthogonal polynomials: symbolic computations, verifications and demonstrations in the Mathematica language. Numer. Algor. 63: 507–520. [15] Olver, S., Townsend, A. (2013). A fast and well-conditioned spectral method. SIAM Rev. 55: 462–489. [16] Szeg¨o, G. (1975). Orthogonal Polynomials. Providence, Rhode Island: American Mathematical Society, Fourth ed. [17] Tao, T. (2019). Conversions between standard polynomial bases. Avail- able at: https://terrytao.wordpress.com/2019/04/07/conversions-between- standard-polynomial-bases/

20 [18] Wikipedia contributors. (2021) “Hermite polynomials.” Available at: https://en.wikipedia.org/wiki/Hermite polynomials [19] Weisstein, E.W. (2021). “Legendre Polynomial.” From MathWorld–A Wol- fram Web Resource. Equation (15) by R. Schmied, 2005. Available at: https://mathworld.wolfram.com/LegendrePolynomial.html [20] Wolfram, D. A. (2021). The change of basis groupoid. arXiv e-print 2107.05450. 23 pp.

A Coefficient Functions

We summarize here coefficient functions based on the results above or derived from formulas in the literature. It has three parts: coefficient functions for change of basis for shifted monomials; for the classical orthogonal polynomials to the monomials; and the monomials to the classical orthogonal polynomials.

A.1 Shifted Monomials The mapping to the monomials from the basis 1, cx+d, (cx+d)2,..., (cx+d)n is given in the first table below. The second table{ is for the inverse mapping. }

Domain Basis Equation α(n,k,c,d)

n n n−k k (cx + d) (11) k c d
Table 3: Mapping to the Monomials

Range Basis Equation α(n,k,c,d)

(cx + d)n (12) n c−n( d)k k −
Table 4: Mapping from the Monomials

21 A.2 Mappings to and from the Monomials for Classical Orthogonal Polynomials

Domain Basis Equation β(n, k)

(λ) k n−2k Γ(n−k+λ) Cn (x) [1], 22.3.4 ( 1) 2 − Γ(λ)k!(n−2k)!

T0(x) [1], 22.4.4 1 k n−2k n (n−k−1)! Tn(x) (n> 0) [1], 22.3.6 ( 1) 2 − 2 k!(n−2k)!

k n−2k n−k Un(x) [1], 22.3.7 ( 1) 2 − k k −n n 2n−
2k Pn(x) [1], 22.3.8 ( 1) 2 − k n k n−2k
n!
Hn(x) [1], 22.3.10 ( 1) 2 − k!(n−2k)!

k −k n! Hen(x) [1], 22.3.11 ( 1) 2 − k!(n−2k)!

Table 5: Mappings to the Monomials using β

22 Domain Basis Equation α(n, k)

n−k (α,β) (−1) Γ(α+n+1) Pn (x) (16) n!Γ(α+β+n+1) k l−n n−l n Γ(α+β+2n−l+1) ( 2) n−k n−l Γ(α+n−l+1) l=0 − P

n k k 1 2 (−1) l (1+n)n−l( 2 −n)l Vn(x) (23) 2n (n−k)! 2 (k−l)! l! ( n ) l=0 P n k k 1 2 (−1) l (1+n)n−l(− 2 −n)l Wn(x) (25) 2n (n−k)! 2 (k−l)! l! ( n ) l=0 P ∗ T0 (x) (32) 1 k ⌊ 2 ⌋ ∗ n n−2v k−v 2(n−v)−k (n−v−1)! Tn (x) (n> 0) (32) 2 k−2v ( 1) 2 v!(n−2v)! v=0 − P
k ⌊ 2 ⌋ ∗ n−2v n−v k−v 2(n−v)−k Un(x) (33) k−2v v ( 1) 2 v=0 − P

2n−k k v 1 ∗ k 2 l (1+n)n−l( 2 −n)l V (x) (34) ( 1) 2n 2 n (n−k)! (k−v)! (v−l)! l! − ( n ) v=0 l=0 P P 2n−k k v 1 ∗ k 2 l (1+n)n−l(− 2 −n)l W (x) (35) ( 1) 2n 2 n (n−k)! (k−v)! (v−l)! l! − ( n ) v=0 l=0 P P k ⌊ 2 ⌋ ∗ −k n−2v 2n−2v n k−v Pn (x) (36) 2 k−2v n v ( 1) v=0 − P


n−k (α) (−1) n+α Ln (x) [1], 22.3.9 (n−k)! k
Table 6: Mappings to the Monomials using α

23 Range Basis Equation β(n, k)

C(λ) x −n λ+n−2k n! n ( ) [10], 18.18.17 2 λ (λ+1)n−kk!

1−n n −n n Tn(x) [12] (2.14) 2 k if 2k

−n
n!
Un(x) [10], 18.18.17 2 (n 2k + 1) − (2)n−kk!

−n n! Pn(x) [10], 18.18.17 2 (2(n 2k)+1) 3 ( )n−kk! − 2

−n n! Hn(x) [10], 18.18.20 2 (n−2k)!k!

−k n! Hen(x) [18] 2 (n−2k)!k!

Table 7: Mappings from the Monomials using β

24 Range Basis Equation α(n, k)

(α,β) Pn (x) (19) (α + β + 2(n k) + 1)(α + β + 2)n−k−1 k − n−l n l (β+1+n−k)k−l (k−l+1)n−k 2 n−l ( 1) (α+β+2)n−l (n−l+α+β+2)n−k l=0 − P
k l (−1) (2(n−l)+1)!! Vn(x) (28) n! (k−l)! (2n−l−k+1)! l! l=0 P k l (−1) (2(n−l)−1)!! Wn(x) (30) (2(n k)+1)n! (k−l)! (2n−l−k+1)! l! − l=0 P k ∗ 1−2n n−v n v Tn (x) (38) 2 k−v v 2 if k

k v ∗ −2n 2 Un(x) (41) 2 (n k + 1)n! k−v (2)n− v+k v!( 2 )! − v=0 2 k−v even P

k k−v l ∗ −n (−1) (2(n−v−l)+1)!! Vn (x) (42) 2 n! (k−v−l)! (2n−v−l−k+1)! v! l! v=0 l=0 P P k k−v l ∗ −n (−1) (2(n−v−l)−1)!! Wn (x) (43) 2 (2(n k)+1)n! (k−v−l)! (2n−v−l−k+1)! v! l! − v=0 l=0 P P k v ∗ −2n 2 Pn (x) (45) 2 (2(n k)+1)n! 3 k−v ( 2 )n− v+k v!( 2 )! − v=0 2 k−v even P (α) Ln (x) [10], 18.18.19 ( n)n k(α + n k + 1)k − − −

Table 8: Mappings from the Monomials using α

25