Download Orthogonal Polynomials and Special Functions: Computation by Walter Gautschi (auth.), Francisco Marcellán, Walter Van PDF

By Walter Gautschi (auth.), Francisco Marcellán, Walter Van Assche (eds.)

Special features and orthogonal polynomials particularly were round for hundreds of years. are you able to think arithmetic with out trigonometric capabilities, the exponential functionality or polynomials? within the 20th century the emphasis used to be on particular features gratifying linear differential equations, yet this has now been prolonged to distinction equations, partial differential equations and non-linear differential equations.

The current set of lecture notes containes seven chapters concerning the present country of orthogonal polynomials and particular services and offers a view on open difficulties and destiny instructions. the themes are: computational tools and software program for quadrature and approximation, equilibrium difficulties in logarithmic power idea, discrete orthogonal polynomials and convergence of Krylov subspace tools in numerical linear algebra, orthogonal rational features and matrix orthogonal rational services, orthogonal polynomials in different variables (Jack polynomials) and separation of variables, a class of finite households of orthogonal polynomials in Askey’s scheme utilizing Leonard pairs, and non-linear unique services linked to the Painlevé equations.

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1 (t) = 0, π0 (t) = 1 be the recurrence relation for {πk ( · ; dλ)}, and let αk± , βk± be the recurrence coefficients for {πk± }. Show that 26 Walter Gautschi β1 = α0+ ⎫ β2k = βk+ /β2k−1 ⎬ k = 1, 2, 3, . . β2k+1 = αk+ − β2k ⎭ (c) Derive relations similar to those in (b) which involve α0+ and αk− , βk− . (d) Write a Matlab program that checks the numerical stability of the nonlinear recursions in (b) and (c) when {πk } are the monic Legendre polynomials. 8. The recurrence relation, in Matlab, of the Chebyshev polynomials of the second kind.

K, where 1 ≤ r1 < r2 < · · · < rK ≤ s, which are atomic measures located at the points ck and having masses mk . m to compute the recurrence matrix B of the special Sobolev orthogonal polynomials. m to check Tables 2–4 in [8], relating to the Hermite measure dλ0 (t) = exp(−t2 )dt and a single atomic measure involving the rth derivative. In the cited reference, the results were obtained by a different method. (c) In the case of the Laguerre measure dλ0 (t) = exp(−t)dt on R+ and rk = k, ck = 0, mk = 1, it may be conjectured that any complex zero that occurs has negative real part.

50) −ρn−1 (z) where now rn = ρn+1 (z) . 51) Computational Methods 23 Similarly as before, we express tˆ πk (t) in two different ways as a linear combination of πk+1 , πk , . . , πk−2 and compare coefficients. By convention, βˆ0 = ˆ = dλ(t) R R dλ(t) = −ρ0 (z). t−z The result is: Algorithm 5 Modification by a linear divisor initialization: α ˆ 0 = α0 + r0 , βˆ0 = −ρ0 (z). continuation (if n > 1): for k = 1, 2, . . , n − 1 do α ˆ k = αk + rk − rk−1 , βˆk = βk−1 rk−1 /rk−2 . Note that here no coefficient αk , βk beyond k ≤ n − 1 is needed, not even βn−1 .

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