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Nonlinear Integral Equation Formulation of Orthogonal Polynomials

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Nonlinear Integral Equation Formulation of Orthogonal Polynomials Nonlinear Integral Equation Formulation of Orthogonal Polynomials Eli Ben-Naim Theory Division, Los Alamos National Laboratory with: Carl Bender (Washington University, St. Louis) C.M. Bender and E. Ben-Naim, J. Phys. A: Math. Theor. 40, F9 (2007) Talk, paper available from: http://cnls.lanl.gov/~ebn BenderFest, St. Louis, MO, March 27, 2009 Working with Carl • Data: 50 email messages received during collaboration • AM: 25%, PM: 75% • “Midnight singularity” 12 10 8 6 Hong Kong 11 4 6 2 4 3 3 3 3 3 # of email messages 2 2 2 2 2 0 1 1 1 1 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 AM PM time of day Orthogonal Polynomials 101 • Definition: A set of polynomials P n ( x ) is orthogonal with respect to the measure g ( x ) on the interval [ α : β ] if β dx g(x)P (x)P (x) = 0 for all m = n n m ! !α • Properties: Orthogonal polynomials have many fascinating and useful properties: ‣ All roots are real and are inside the interval [α : β] ‣ The orthogonal polynomials form a complete basis: any polynomial is a linear combination of orthogonal polynomials of lesser or equal order Orthogonal Polynomials 101 • Specification: Orthogonal polynomials can be specified in multiple ways (typically linear) Example: Legendre Polynomials orthogonal on [ 1 : 1] w.r.t g(x) = 1 − 1. Differential Equation: 2 (1 x )P !!(x) 2xP ! (x) + n(n + 1)P (x) = 0 − n − n n 2. Generating Function: 1 ∞ n = Pn(x)t √ 2 1 2tx + t n=0 Rodriguez Formula: − ! 3. 1 dn P (x) = (x2 1)n n 2n n! dxn − 4. Recursion Formula: (n + 1)Pn+1(x) (2n + 1)xPn(x) + nPn 1(x) = 0 − − Nonlinear Integral Equation • Consider the following nonlinear integral equation β P (x) = dy w(y) P (y) P (x + y) !α • No restriction on integration limits • Weight function restricted: non-vanishing integral β dx w(x) = 0 ! !α • Motivation: stochastic process involving subtraction x ∞ y (x+y) x1, x2 x1 x2 e− = 2 dy e− e− → | − | !0 • Reduces to the Wigner function equation when [α : β] = [ : ] w(y) = eiy −∞ ∞ Constant Solution (n=0) • Consider the constant polynomial P (x) = a a = 0 0 ! • A constant solves the nonlinear integral equation β P (x) = dy w(y) P (y) P (x + y) α When ! • β a = dy w(y)a2 !α • Since we can divide by a β 1 = dy w(y)a. !α A constant solution exists Linear Solution (n=1) • Consider the linear polynomial P1(x) = a + bx b = 0 ! Let us introduce the shorthand notation • β f(x) dx w(x)f(x) ! " ≡ !α The nonlinear integral equation P ( x ) = P (y ) P ( x + y ) reads • ! " a + bx = P (y) [a + by + bx] ! 1 " 1. Equate the coefficients of x 1 and divide by b = 0 ! b = b P P (x) = 1 ! 1" ⇒ ! 1 " 2. Equate the coefficients of x 0 and divide by a = a P + b yP (y) xP (x) = 0 ! 1" ! 1 " ⇒ ! 1 " Linear solution generally exists Nonlinear equation reduces to 2 linear inhomogeneous equations for a,b Quadratic Solution (n=2) • Consider the quadratic polynomial P (x) = a + bx + cx2 c = 0 2 ! • The nonlinear integral equation becomes a + bx + cx2 = P (y) a + by + cy2 + bx + 2cxy + cx2 ! 2 " • Successively equating! coefficients " c = c P (y) P (x) = 1 ! 2 " ⇒ ! 2 " b = b P (y) + 2c yP (y) xP (x) = 0 ! 2 " ! 2 " ⇒ ! 2 " a = a P (y) + b yP (y) + c y2P (y) x2P (x) = 0 ! 2 " ! 2 " ! 2 " ⇒ ! 2 " Quadratic solution generally exists Miraculous cancelation of terms Nonlinear equation reduces to 3 linear inhomogeneous equations for a,b,c General Properties • The nonlinear integral equation has two remarkable properties: 1. This equation preserves the order of a polynomial 2. For polynomial solutions, the nonlinear equation reduces to a linear set of equations for the coefficients of the polynomials • In general, a polynomial of degree n n k Pn(x) = an,k x k=0 • Is a solution of the integral equation! if and only if its n+1 coefficients satisfy the following set of n+1 linear inhomogeneous equations xkP (x) = δ (k = 0, 1, . , n) ! n " k,0 Infinite number of polynomial solutions The set of polynomial solutions is orthogonal! • The polynomial solutions are orthogonal w.r.t. g(x) = x w(x) Because for m < n • m m+1 k+1 k x PnPm = am,k x Pn(x) = am,k 1 x Pn(x) = 0 ! " ! " − ! " k!=0 k!=1 • As follows immediately from xkP (x) = δ ! n " k,0 1.The nonlinear integral equation admits an infinite set of polynomial solutions 2.The polynomial solutions form an orthogonal set The equations for the coefficients • The equations for the coefficients xkP (x) = δ (k = 0, 1, . , n) ! n " k,0 Can be compactly written as • n an,jmk+j = δk,0 (k = 0, 1, . , n) j=0 • Or in! matrix form m m m a 1 0 1 · · · n n,0 m1 m2 mn+1 an,1 0 . ·.· · . . = . . .. m m m a 0 n n+1 · · · 2n n,n • In terms of the “moments” of the weight function m = xn n ! " Formulas for the polynomials • Using Cramer’s rule, the polynomials can be expressed as a ratio of determinants n 1 x x m0 m1 mn m m · · · m m m · · · m 1 2 · · · n+1 1 2 · · · n+1 An = . , Bn = . . .. .. m m m m m m n n+1 · · · 2n n n+1 · · · 2n det An Pn(x) = det Bn • Explicit expressions P0(x) = 1, m xm P (x) = 2 − 1 , 1 m m2 2 − 1 (m m m2) + (m m m m )x + (m m m2)x2 P (x) = 2 4 − 3 2 3 − 1 4 1 3 − 2 . 2 m (m m2) m2 + 2m m m m3 4 2 − 1 − 3 1 2 3 − 2 Examples • Interval [0:1], weight function w(x)=1 P0(x) = 1 P (x) = 4 6x 1 − P (x) = 9 36x + 30x2 2 − P (x) = 16 120x + 240x2 140x3. 3 − − Jacobi polynomials P ( x ) G (2 , 2 , x ) orthogonal • n ∝ n w.r.t the measure g(x)=x 1. Generalized Laguerre polynomials (γ) γ 1 x L (x) α = 0, β = , w(x) = x − e− n ∞ 2. Jacoby polynomials q 2 p q G (p, q, x) α = 0, β = 1, w(x) = x − (1 x) − n − 3. Shifted Chebyshev of the second kind polynomials 1/2 1/2 U (x) α = 0, β = 1, w(x) = (1 x) x− n∗ − Integration in the complex domain • To specify the Legendre Polynomials 1 1 P (x) = dy P (y) P (x + y) 1 y !− • Perform integration in the complex domain 1 1 dx w(x) = = iπ iπ x 1 x !− • This integration path gives the Legendre Polynomials P0(x) = 1, iπ P (x) = x, 1 2 P (x) = 1 3x2, 2 − 3iπ P (x) = (3x 5x3), 3 8 − Nonlinear integral equation extends to complex domain Generalization I: multiplicative arguments Nonlinear equation with multiplicative argument • β P (x) = dy w(y) P (y) P (x y) !α • Infinite set of polynomial solutions when xkP (x) = 1 ! n " These polynomials are orthogonal • m (1 x)P P = a xkP xk+1P = 0 m < n ! − n m# m,k ! n# − ! n# k=0 • Now, the orthog!onality measur" e is # g(x) = (1 x)w(x) − A series of nonlinear integral formulations Generalization II: iterated integrals • Iterate the nonlinear integral equation β g(y) P (x) = dy P (y) P (x + y) y !α • The double integral equation β g(y) β g(z) P (x) = dy dz P (y) P (z) P (x + y + z) y z !α !α Similarly specifies orthogonal polynomials Summary A set of orthogonalβ polgynomials(y) w.r.t the measure g(x) • P (x) = dy P (y) P (x + y) can be specified through ythe nonlinear integral equation !α • For polynomial solutions, this nonlinear equation reduces to a linear set of equations • Simple, compact, and completely general way to specify orthogonal polynomials • Natural way to extend theory to complex domain Outlook • Non-polynomial solutions • Multi-dimensional polynomials • Matrix polynomials • Polynomials defined on disconnected domains • Higher-order nonlinear integral equations • Use nonlinear formulation to derive integral identities • Asymptotic properties of polynomials Nonlinear Integral Identities • Let P n ( x ) be the set of orthogonal polynomials specified by the nonlinear integral equation β g(y) P (x) = dy P (y) P (x + y) n y n n !α • Then, any polynomial Q m ( x ) of degree m n satisfies the nonlinear integral identity ≤ β g(y) Q (x) = dy P (y) Q (x + y) m y n m !α Asymptotic Properties & Integral Identities • Combining the asymptotics of the Laguerre Polynomials γ γ x γ/2 lim n− Ln = x− Jγ (2√x) n n →∞ ! " • And the nonlinear integral equation with scaled variables 1 γ x 1 ∞ γ 1 y/n 1 γ y 1 γ x + y γ Ln = dy y − e− γ Ln γ Ln n n Γ(γ) 0 n n n n ! " # ! " $ % • Gives a standard identity for the Bessel functions 2 2 γ 1 Jγ (z) 1 ∞ γ 1 Jγ (√w + z ) 2 − = dw w − J (w) 2zγ Γ(γ) γ (w2 + z2)γ/2 !0.
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