An overview of Classical Orthogonal Polynomials Roberto S. Costas-Santos University of Alcal´ a Work supported by MCeI grant MTM2009-12740-C03-01 www.rscosan.com Gaithersburg, March 25, 2014 NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
Outline 1 The basics Classical Orthogonal Polynomials The Favard’s theorem My First result: the Degenerate Favard’s Theorem 2 The Schemes The Classical Hypergeometric Orthogonal Polynomials The Classical basic Hypergeometric Orth. Polyn. 3 Some Results Characterization Theorem Hypergeometric and basic hypergeometric representations The Connection Problem One example. Big q -Jacobi polynomials NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
THE BASICS NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
Classical Orthogonal Polynomials Let ( P n ) be a polynomial sequence and u be a functional. Property of orthogonality � u , P n P m � = d 2 n δ n , m . Distributional equation: D ( φ u ) = ψ u , deg ψ ≥ 1 , deg φ ≤ 2 . Three-term recurrence relation: xP n ( x ) = α n P n +1 ( x ) + β n P n ( x ) + γ n P n +1 ( x ) . The weight function d µ ( z ) = ω ( z ) dz � � u , P � = P ( z ) d µ ( z ) , Γ ⊂ C , . Γ NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
1 Continuous classical orthogonal polynomials d dx ( φ ( x ) ω ( x )) = ψ ( x ) ω ( x ), 2 ∆-classical orthogonal polynomials ∇ ( φ ( x ) ω ( x )) = ψ ( x ) ω ( x ), ∆ f ( x ) = f ( x + 1) − f ( x ), ∇ f ( x ) = f ( x ) − f ( x − 1), 3 q -Hahn classical orthogonal polynomials D 1 / q ( φ ( x ) ω ( x )) = ψ ( x ) ω ( x ), D q f ( x ) = f ( qx ) − f ( x ) x ( q − 1) , x � = 0, D q f (0) = f ′ (0), x ( s ) = c 1 q s + c 2 . NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
Some families Continuous Classical OP: Jacobi, Hermite, Laguerre and Bessel. ∆-Classical OP: Hahn, Racah, Meixner, Krawtchouk, Charlier, etc. q -Classical OP: Askey Wilson, q -Racah, q -Hahn, Continuous q -Hahn, Big q -Jacobi, q -Hermite, q -Laguerre, Al-Salam-Chihara, Stieltjes-Wigert, etc. NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
Some families Continuous Classical OP: Jacobi, Hermite, Laguerre and Bessel. ∆-Classical OP: Hahn, Racah, Meixner, Krawtchouk, Charlier, etc. q -Classical OP: Askey Wilson, q -Racah, q -Hahn, Continuous q -Hahn, Big q -Jacobi, q -Hermite, q -Laguerre, Al-Salam-Chihara, Stieltjes-Wigert, etc. NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
Some families Continuous Classical OP: Jacobi, Hermite, Laguerre and Bessel. ∆-Classical OP: Hahn, Racah, Meixner, Krawtchouk, Charlier, etc. q -Classical OP: Askey Wilson, q -Racah, q -Hahn, Continuous q -Hahn, Big q -Jacobi, q -Hermite, q -Laguerre, Al-Salam-Chihara, Stieltjes-Wigert, etc. NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
The Favard’s theorem Let ( p n ) n ∈ N 0 generated by the TTRR xp n ( x ) = p n +1 ( x ) + β n p n ( x ) + γ n p n − 1 ( x ) . Favard’s theorem If γ n � = 0 ∀ n ∈ N then there exists a moments functional L 0 : P [ x ] → C so that L 0 ( p n p m ) = r n δ n , m with r n a non-vanishing normalization factor. NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
Degenerate version of Favard’s theorem Theorem If there exists N so that γ N = 0, then ( p n ) is a MOPS with respect to � L 1 ( T ( N ) ( f ) T ( N ) ( g )) . � f , g � = L 0 ( fg ) + j ∈ A NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
THE RELEVANT FAMILIES NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
The Classical Hypergeometric Orthogonal Polynomials Wilson Racah F Cont. Dual Hahn Cont. Hahn Hahn Dual Hahn F F Krawchuk Meixner-Pollaczek Jacobi Meixner F laguerre Charlier Hermite NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
The Classical basic Hypergeometric Orth. Polyn. The scheme is too big to put it on here, let’s go outside to see it ;) NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
SOME RESULTS NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
Characterization Theorems. The continuous version Let ( P n ) be an OPS with respect to ω . The following statements are equivalent: 1 P n is classical, i.e. ( φ ( x ) ω ( x )) ′ = ψ ( x ) ω ( x ). 2 ( P ′ n +1 ) is a OPS. 3 ( P ( k ) n + k ) is a OPS for any integer k . 4 (First structure relation) α n P n +1 ( x ) + � φ ( x ) P ′ n ( x ) = � β n P n ( x ) + � γ n P n − 1 ( x ) . 5 (Second structure relation) n +1 ( x ) + � α n P ′ β n P ′ γ n P ′ P n ( x ) = � n ( x ) + � n − 1 ( x ) . 6 (Eigenfunctions of SODE) φ ( x ) P ′′ ( x ) + ψ ( x ) P ′ ( x ) + λ P ( x ) = 0 . NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
Characterization Theorem (cont.) Let ( P n ) be an OPS with respect to ω . The following statements are equivalent: 1 P n is classical, i.e. ( φ ( x ) ω ( x )) ′ = ψ ( x ) ω ( x ). 2 The Rodrigues Formula for P n d n � B n φ n ( x ) ω ( x )) , P n ( x ) = B n � = 0 . dx n ω ( x ) 3 φ ( x )( P n P n − 1 ) ′ ( x ) = g n P 2 n ( x ) − ( ψ ( x ) − φ ′ ( x )) P n ( x ) P n − 1 ( x ) + h n P 2 n − 1 ( x ) NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
Hypergeometric and basic hypergeometric representations The continuous and discrete COP can be written in terms of � a 1 , a 2 , . . . , a r � � � � z k ( a 1 ) k ( a 2 ) k . . . ( a r ) k � r F s � z = k ! . b 1 , b 2 , . . . , b s ( b 1 ) k ( b 2 ) k . . . ( b s ) k k ≥ 0 The q -discrete COP can be written in terms of � � a 1 , . . . , a r � � 2 ) � 1+ s − r � � z k ( a 1 ; q ) k . . . ( a r ; q ) k ( − 1) k q ( k � r ϕ s � z = . b 1 , . . . , b s ( b 1 ; q ) k . . . ( b s ; q ) k ( q ; q ) k k ≥ 0 ( a ) k = a ( a + 1) · · · ( a + k − 1) ( a ; q ) k = (1 − a )(1 − aq ) · · · (1 − aq k − 1 ) NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
The Connection Problem The connection problem is the problem of finding the coefficients c k ; n in the expansion of P n in terms of another sequence of polynomials R k , i.e. n � P n ( x ) = c k ; n R k ( x ) . k =0 We are interested into obtaining such coefficients for Classical orthogonal polynomials in a enough ‘general’ context. NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
The example. Big q -Jacobi polynomials Again let’s go to File 2 :D NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
Some References (with J.F. S´ anchez-Lara) Extensions of discrete classical orthogonal polynomials beyond the orthogonality. J. Comput. Appl. Math. 225 (2009), no. 2, 440–451 (with F. Marcell´ an) q -Classical orthogonal polynomial: A general difference calculus approach. Acta Appl. Math. 111 (2010), no. 1, 107–128 (with J.F. S´ anchez-Lara) Orthogonality of q -polynomials for non-standard parameters. J. Approx. Theory 163 (2011), no. 9, 1246–1268 (with F. Marcell´ an) The complementary polynomials and the Rodrigues operator of classical orthogonal polynomials. Proc. Amer. Math. Soc. 140 (2012), no. 10, 3485–3493 NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
FINALLY.... THANK YOU FOR YOUR ATTENTION !! NIST, 2014 R. S. Costas-Santos : An overview of Classical Orthogonal Polynomials
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