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Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Generalized Fourier Series for Solutions of Linear Differential Equations Alexandre Benoit, Joint work with Bruno Salvy INRIA (France) February 15, 2011 1 /

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1 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Generalized Fourier Series for Solutions of Linear Differential Equations

Alexandre Benoit,

Joint work with Bruno Salvy

INRIA (France)

February 15, 2011

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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2 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

I Introduction

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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3 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Generalized Fourier Series

f (x) =

  • anψn(x)

Some Examples sin(x) = 2

∞

  • n=0

(−1)nJ2n (x) arccos (x) = 1 2π T0 (x) −

∞

  • n=0

4 (2 n + 1)2 π T2n+1 (x) erf (x) = 2

∞

  • n=0
  • −1

4 n 1 √π (2 n + 1) n! 1F1 n + 1

2

2n + 2

  • − x
  • More generally (ψn(x))n∈N can be an orthogonal basis of a Hilbert space.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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4 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Applications: Good approximation properties.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

x Approximation of arctan(2x) by Taylor expansion of degree 1 −0.5 −0.25 0.25 0.5 −1 −0.5 0.5 1 arctan(2x) Taylor approximation

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4 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Applications: Good approximation properties.

x Bad approximation outside its circle of convergence −1 −0.5 0.5 1 −1.5 −1 −0.5 0.5 1 1.5 arctan(2x) Taylor approximation

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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4 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Applications: Good approximation properties.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

x approximation of arctan(2x) by Chebyshev expansion of degree 1 −1 −0.5 0.5 1 −1.5 −1 −0.5 0.5 1 1.5 arctan(2x) Chebyshev expansion Taylor expansion

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4 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Applications: Good approximation properties.

x bad approximation over R −2 −1 1 2 −1.5 −1 −0.5 0.5 1 1.5 Taylor expansion Chebyshev expansion arctan(2x)

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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4 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Applications: Good approximation properties.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

x approximation of arctan(2x) by Hermite expansion of degree 1 −2 −1 1 2 −1.5 −1 −0.5 0.5 1 1.5 Taylor expansion Chebyshev expansion arctan(2x) Hermite expansion

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5 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Our framework

Families of functions ψn(x) with two special properties Mult by x (Px) Recx2 (xψn(x)) = Recx1 (ψn(x)) Examples Monomial polynomials (Mn = xn) All orthogonal polynomials Bessel functions Legendre functions Parabolic cylinder functions

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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5 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Our framework

Families of functions ψn(x) with two special properties Mult by x (Px) Recx2 (xψn(x)) = Recx1 (ψn(x)) Examples Monomial polynomials (Mn = xn) All orthogonal polynomials Bessel functions Legendre functions Parabolic cylinder functions xMn = Mn+1 2xTn(x) = Tn+1(x) + Tn−1(x) 1 n (xJn+1 − xJn−1) = 2Jn

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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5 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Our framework

Families of functions ψn(x) with two special properties Mult by x (Px) Recx2 (xψn(x)) = Recx1 (ψn(x)) Differentiation (P∂) Rec∂2 (ψ′

n(x)) = Rec∂1 (ψn(x))

Examples

Monomial polynomials Classical orthogonal polynomials Bessel functions Legendre functions Parabolic cylinder functions

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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5 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Our framework

Families of functions ψn(x) with two special properties Mult by x (Px) Recx2 (xψn(x)) = Recx1 (ψn(x)) Differentiation (P∂) Rec∂2 (ψ′

n(x)) = Rec∂1 (ψn(x))

Examples

Monomial polynomials Classical orthogonal polynomials Bessel functions Legendre functions Parabolic cylinder functions M′

n = nMn−1

1 n + 1T ′

n+1(x) −

1 n − 1T ′

n−1(x) = 2Tn(x)

2J′

n(x) = Jn−1(x) − Jn+1(x)

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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5 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Our framework

Families of functions ψn(x) with two special properties Mult by x (Px) Recx2 (xψn(x)) = Recx1 (ψn(x)) Differentiation (P∂) Rec∂2 (ψ′

n(x)) = Rec∂1 (ψn(x))

This is our data-structure for ψn(x)

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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6 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Main Idea

Main Idea If ψn(x) satisfies (Px) and (P∂), for any f (x) = anψn(x) solution of a linear differential equation with polynomial coefficients, the coefficients an are cancelled by a linear recurrence relation with polynomial coefficients.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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6 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Main Idea

Main Idea If ψn(x) satisfies (Px) and (P∂), for any f (x) = anψn(x) solution of a linear differential equation with polynomial coefficients, the coefficients an are cancelled by a linear recurrence relation with polynomial coefficients. Applications: Efficient numerical computation of the coefficients. Computation of closed-form for the coefficients (when it’s possible).

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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7 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Previous work

Clenshaw (1957): numerical scheme to compute the coefficients when ψn(x) = Tn(x) (Chebyshev series). Lewanowicz (1976-2004): algorithms to compute a recurrence relation when ψn is an orthogonal or semi-orthogonal polynomial family. Rebillard and Zakrajˇ sek (2006): General algorithm computing a recurrence relation when ψn is a family of hypergeometric polynomials Benoit and Salvy (2009) : Simple unified presentation and complexity analysis of the previous algorithms using Fractions of recurrence relations when ψn = Tn. New and fast algorithm to compute the Chebyshev recurrence.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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8 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

New Results (2011)

Simple unified presentation of the previous algorithms using Pairs of recurrence relations. New general algorithm computing the recurrence relation of the coefficients for a Generalized Fourier Series when ψn(x) satisfies (Px) and (P∂).

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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9 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

II Pairs of Recurrence Relations

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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10 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Examples: Chebyshev case (f (x) = unTn(x))

Basic rules: xf =

  • anTn

(Px) − − → an = un−1 + un+1 2 f ′ =

  • bnTn

(P∂) − − → bn−1 − bn+1 = 2nun.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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10 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Examples: Chebyshev case (f (x) = unTn(x))

Basic rules: xf =

  • anTn

(Px) − − → an = un−1 + un+1 2 f ′ =

  • bnTn

(P∂) − − → bn−1 − bn+1 = 2nun. Combine: f ′ + 2xf =

  • cnTn

(P∂ + 2Px) − − − − − − − → cn−1 − cn+1 = Rec1(un). Application: Chebyshev series for exp(−x2).

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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10 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Examples: Chebyshev case (f (x) = unTn(x))

Basic rules: xf =

  • anTn

(Px) − − → an = un−1 + un+1 2 f ′ =

  • bnTn

(P∂) − − → bn−1 − bn+1 = 2nun. Combine: f ′ + 2xf =

  • cnTn

(P∂ + 2Px) − − − − − − − → cn−1 − cn+1 = Rec1(un). Application: Chebyshev series for exp(−x2). (f ′ + 2xf )′ =

  • dnTn

(P∂) − − → dn−1 − dn+1 = 2ncn, → Rec2(dn) = Rec3(un), (f ′ + 2xf )′ − 2f =

  • enTn

→ Rec4(en) = Rec5(un). Application: Chebyshev series for erf(x).

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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11 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Rings of Pairs of Recurrence Relations

Theorem (Least Common Left Multiple (Ore 33)) Given Rec1 and Rec2, there exists a recurrence relation Rec and a pair

  • Rec1,

Rec2

  • such that for all sequences (un)n∈N :

Rec (un) = Rec1 ◦ Rec1 (un) = Rec2 ◦ Rec2 (un)

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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11 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Rings of Pairs of Recurrence Relations

Theorem (Least Common Left Multiple (Ore 33)) Given Rec1 and Rec2, there exists a recurrence relation Rec and a pair

  • Rec1,

Rec2

  • such that for all sequences (un)n∈N :

Rec (un) = Rec1 ◦ Rec1 (un) = Rec2 ◦ Rec2 (un) The LCLM is the recurrence relation Rec with minimal order.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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11 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Rings of Pairs of Recurrence Relations

Theorem (Least Common Left Multiple (Ore 33)) Given Rec1 and Rec2, there exists a recurrence relation Rec and a pair

  • Rec1,

Rec2

  • such that for all sequences (un)n∈N :

Rec (un) = Rec1 ◦ Rec1 (un) = Rec2 ◦ Rec2 (un) The LCLM is the recurrence relation Rec with minimal order. Computation : Euclidean algorithm.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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12 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Operations of addition and composition

Rec = lclm(Rec1, Rec2) = Rec1 ◦ Rec1 = Rec2 ◦ Rec2 Operation 1: Addition Rec1(an) = Rec3(un), Rec2(bn) = Rec4(un) Rec(an) = Rec1 ◦ Rec3(un), Rec(bn) = Rec2 ◦ Rec4(un) → Rec(an + bn) =

  • Rec1 ◦ Rec3 +

Rec2 ◦ Rec4

  • (un).

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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12 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Operations of addition and composition

Rec = lclm(Rec1, Rec2) = Rec1 ◦ Rec1 = Rec2 ◦ Rec2 Operation 1: Addition Rec1(an) = Rec3(un), Rec2(bn) = Rec4(un) Rec(an) = Rec1 ◦ Rec3(un), Rec(bn) = Rec2 ◦ Rec4(un) → Rec(an + bn) =

  • Rec1 ◦ Rec3 +

Rec2 ◦ Rec4

  • (un).

Operation 2: Composition Rec1(un) = Rec3(an), Rec2(un) = Rec4(bn) Rec(un) = Rec1 ◦ Rec1(un) = Rec2 ◦ Rec2(un) → Rec1 ◦ Rec3(an) = Rec2 ◦ Rec4(bn).

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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13 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Main Result

Main Result : Morphism There exists a morphism ϕ such that if f = unψn(x) and g = vnψn(x) are related by L (f ) = g (L a linear differential operator), then: ϕ (L) = (Rec1, Rec2) with Rec1 (un) = Rec2 (vn) In particular if L (f ) = 0, then Rec1 (un) = 0.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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14 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Definition of the Morphism ϕ

f =

  • unψn(x)

g =

  • vnψn(x)

Recx2 (xψn(x)) = Recx1 (ψn(x)) if xf = g, then Recx2 (un) = Recx1 (vn) ϕ(x) Rec∂2 (ψ′

n(x)) = Rec∂1 (ψn(x))

if f ′ = g, then Rec∂1 (un) = Rec∂2 (vn) ϕ(∂)

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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14 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Definition of the Morphism ϕ

f =

  • unψn(x)

g =

  • vnψn(x)

Recx2 (xψn(x)) = Recx1 (ψn(x)) if xf = g, then Recx2 (un) = Recx1 (vn) ϕ(x) Rec∂2 (ψ′

n(x)) = Rec∂1 (ψn(x))

if f ′ = g, then Rec∂1 (un) = Rec∂2 (vn) ϕ(∂) Example for Chebyshev series: 2xTn(x) = Tn+1(x) + Tn−1(x) T ′

n+1(x)

n + 1 − T ′

n−1(x)

n − 1 = 2Tn(x) un+1 + un−1 = 2vn 2un = 1 n (vn−1 − vn+1) ϕ Example for Bessel series 1 n (xJn+1 − xJn−1) = 2Jn 2J′

n(x) = Jn−1(x) − Jn+1(x)

2un = vn+1 n + 1 + vn−1 n − 1 un+1 − un−1 = 2vn ϕ

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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15 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

General Algorithm

Recall Definition of ϕ (x) and ϕ (∂) Algorithms to compute addition and composition between two pairs

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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15 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

General Algorithm

Recall Definition of ϕ (x) and ϕ (∂) Algorithms to compute addition and composition between two pairs General Algorithm We deduce from this morphism a general Horner-like algorithm to compute the recurrence relation satisfied by the coefficients of a generalized Fourier series solution of a linear differential equation.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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16 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

III Recurrences of Smaller Order

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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17 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Greatest Common Left Divisor and Reduction of Order

GCLD

Given a pair (Rec1, Rec2), the Euclidean algorithm computes the greatest recur- rence relation Rec (GCLD) such that there exists a pair

  • Rec1,

Rec2

  • with the

following relations for all sequences (un)n∈N and (vn)n∈N:

Rec ◦ Rec1 (un) = Rec1 (un) Rec ◦ Rec2 (vn) = Rec2 (vn)

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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17 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Greatest Common Left Divisor and Reduction of Order

GCLD

Given a pair (Rec1, Rec2), the Euclidean algorithm computes the greatest recur- rence relation Rec (GCLD) such that there exists a pair

  • Rec1,

Rec2

  • with the

following relations for all sequences (un)n∈N and (vn)n∈N:

Rec ◦ Rec1 (un) = Rec1 (un) Rec ◦ Rec2 (vn) = Rec2 (vn) The orders of the recurrence relations Reci are at most those of Reci.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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17 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Greatest Common Left Divisor and Reduction of Order

GCLD

Given a pair (Rec1, Rec2), the Euclidean algorithm computes the greatest recur- rence relation Rec (GCLD) such that there exists a pair

  • Rec1,

Rec2

  • with the

following relations for all sequences (un)n∈N and (vn)n∈N:

Rec ◦ Rec1 (un) = Rec1 (un) Rec ◦ Rec2 (vn) = Rec2 (vn) The orders of the recurrence relations Reci are at most those of Reci. Remark In a general case, we don’t have : Rec1(un) = Rec2(vn) ⇒ Rec1(un) = Rec2(vn),

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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17 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Greatest Common Left Divisor and Reduction of Order

GCLD

Given a pair (Rec1, Rec2), the Euclidean algorithm computes the greatest recur- rence relation Rec (GCLD) such that there exists a pair

  • Rec1,

Rec2

  • with the

following relations for all sequences (un)n∈N and (vn)n∈N:

Rec ◦ Rec1 (un) = Rec1 (un) Rec ◦ Rec2 (vn) = Rec2 (vn) The orders of the recurrence relations Reci are at most those of Reci. Remark In a general case, we don’t have : Rec1(un) = Rec2(vn) ⇒ Rec1(un) = Rec2(vn), (−1)n+2 − (−1)n = (−1)2(n+1) − (−1)2n (−1)n+1 + (−1)n = (−1)2n

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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18 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

GLCD for reduction of order

Theorem Given L a linear differential operator, f = unψn(x), g = vnψn(x) such that L (f ) = g and a pair (Rec1, Rec2) = ϕ(L). We have

  • Rec1 (un) =

Rec2 (vn)

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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18 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

GLCD for reduction of order

Theorem Given L a linear differential operator, f = unψn(x), g = vnψn(x) such that L (f ) = g and a pair (Rec1, Rec2) = ϕ(L). We have

  • Rec1 (un) =

Rec2 (vn) Application: Adaptation of the previous algorithm At the end of the previous algorithm, add a final step: Remove the GCLD of the two recurrence relations of the pair.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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19 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Example of reduction for Chebyshev series

  • 1 − x2 =
  • n∈N

4 π(2n + 1)T2n(x) =

  • n∈N

cnTn(x) √ 1 − x2 is the solution of the differential equation: xy(x) + (1 − x2)y ′(x) = 0

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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19 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Example of reduction for Chebyshev series

  • 1 − x2 =
  • n∈N

4 π(2n + 1)T2n(x) =

  • n∈N

cnTn(x) √ 1 − x2 is the solution of the differential equation: xy(x) + (1 − x2)y ′(x) = 0 With the general algorithm we obtain the pair of recurrence relations : Rec1 (un) = (n+3)un+2−2nun+(n−3)un−2 and Rec2 (vn) = 2 (−vn+1 + vn−1) . We deduce : (n + 3)cn+2 − 2ncn + (n − 3)cn−2 = 0.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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19 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Example of reduction for Chebyshev series

  • 1 − x2 =
  • n∈N

4 π(2n + 1)T2n(x) =

  • n∈N

cnTn(x) √ 1 − x2 is the solution of the differential equation: xy(x) + (1 − x2)y ′(x) = 0 With the general algorithm we obtain the pair of recurrence relations : Rec1 (un) = (n+3)un+2−2nun+(n−3)un−2 and Rec2 (vn) = 2 (−vn+1 + vn−1) . We deduce : (n + 3)cn+2 − 2ncn + (n − 3)cn−2 = 0.

  • Rec1 (un) = (n + 2)un+1 − (n − 2)un−1 and

Rec2 (vn) = 2vn. We deduce : (n + 2)cn+1 − (n − 2)cn−1 = 0.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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20 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

IV Conclusion

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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21 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Conclusion

Contributions: Use of Pairs of recurrence relations. New general algorithm. Use of the GCLD to reduce order of the recurrence.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.

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21 / 21 Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion

Conclusion

Contributions: Use of Pairs of recurrence relations. New general algorithm. Use of the GCLD to reduce order of the recurrence. Perspectives: Computation of the recurrence of minimal order. Numerical computation of the coefficients. Closed form for the coefficients. Example erf (x) =

∞

  • n=0

2 4−n (−1)n

1F1(n + 1/2; 2 n + 2; −1)

√π (2 n + 1) n! T2 n+1 (x) . Integration in the Dynamic Dictionary of Mathematical Functions.

Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.