457 research outputs found
Casoratian identities for the discrete orthogonal polynomials in discrete quantum mechanics with real shifts
No full textIn our previous papers [S. Odake and R. Sasaki, J. Phys. A 46, 245201 (2013) and S. Odake and R. Sasaki, J. Approx. Theory 193, 184 (2015)], the Wronskian identities for the Hermite, Laguerre, and Jacobi polynomials and the Casoratian identities for the Askey-Wilson polynomial and its reduced-form polynomials were presented. These identities are naturally derived through quantum-mechanical formulation of the classical orthogonal polynomials: ordinary quantum mechanics for the former and discrete quantum mechanics with pure imaginary shifts for the latter. In this paper we present the corresponding identities for the discrete quantum mechanics with real shifts. Infinitely many Casoratian identities for the q-Racah polynomial and its reduced-form polynomials are obtained
I remember sports at Bridgeton High School
No full textAlthough Satoru Tsufura only lived at Seabrook for a year, he still contributed to Seabrook's reputation of producing outstanding athletes. In fact, Sat was the only 4-letter athlete in his graduating class at Bridgeton High School. He won a scholarhip in his senior year, and he was chosen as an all-star baseball player and was invited to play in a special baseball game against minor or major league baseball players. The Seabrook Educational and Cultural Center has been soliciting current and past residents of Seabrook Farms for an "I remember" project. Residents are asked to create narratives regarding their experiences at Seabrook Farms. These memories help preserve the history and multi-cultural heritage of Seabrook Farms
Another set of infinitely many exceptional (Xℓ) Laguerre polynomials
No full textAbstractWe present a new set of infinitely many shape invariant potentials and the corresponding exceptional (Xℓ) Laguerre polynomials. They are to supplement the recently derived two sets of infinitely many shape invariant thus exactly solvable potentials in one-dimensional quantum mechanics and the corresponding Xℓ Laguerre and Jacobi polynomials [S. Odake, R. Sasaki, Phys. Lett. B 679 (2009) 414]. The new Xℓ Laguerre polynomials and the potentials are obtained by a simple limiting procedure from the known Xℓ Jacobi polynomials and the potentials, whereas the known Xℓ Laguerre polynomials and the potentials are obtained in the same manner from the mirror image of the known Xℓ Jacobi polynomials and the potentials
Another set of infinitely many exceptional (X(l)) Laguerre polynomials
No full textWe present a new set of infinitely many shape invariant potentials and the corresponding exceptional (X-l) Laguerre polynomials. They are to supplement the recently derived two sets of infinitely many shape invariant thus exactly solvable potentials in one-dimensional quantum mechanics and the corresponding X(l) Laguerre and Jacobi polynomials [S. Odake, R. Sasaki, Phys. Lett. B 679 (2009) 414]. The new X(l) Laguerre polynomials and the potentials are obtained by a simple limiting procedure from the known X(l) Jacobi polynomials and the potentials, whereas the known X(l) Laguerre polynomials and the potentials are obtained in the same manner from the mirror image of the known X(l) Jacobi polynomials and the potentials.ArticlePHYSICS LETTERS B. 684(2-3):173-176 (2010)journal articl
Properties of the Exceptional (X_l) Laguerre and Jacobi Polynomials
No full textWe present various results on the properties of the four infinite sets of the exceptional X_l polynomials discovered recently by Odake and Sasaki [Phys. Lett. B 679 (2009), 414-417; Phys. Lett. B 684 (2010), 173-176]. These Xl polynomials are global solutions of second order Fuchsian differential equations with l+3 regular singularities and their confluent limits. We derive equivalent but much simpler looking forms of the X_l polynomials. The other subjects discussed in detail are: factorisation of the Fuchsian differential operators, shape invariance, the forward and backward shift operations, invariant polynomial subspaces under the Fuchsian differential operators, the Gram-Schmidt orthonormalisation procedure, three term recurrence relations and the generating functions for the X_l polynomials
Properties of the Exceptional (X-l) Laguerre and Jacobi Polynomial
No full textWe present various results on the properties of the four infinite sets of the exceptional Xl polynomials discovered recently by Odake and Sasaki [Phys. Lett. B 679 (2009), 414-417; Phys. Lett. B 684 (2010), 173-176]. These Xl polynomials are global solutions of second order Fuchsian differential equations with l+3 regular singularities and their confluent limits. We derive equivalent but much simpler looking forms of the Xl polynomials. The other subjects discussed in detail are: factorisation of the Fuchsian differential operators, shape invariance, the forward and backward shift operations, invariant polynomial subspaces under the Fuchsian differential operators, the Gram-Schmidt orthonormalisation procedure, three term recurrence relations and the generating functions for the Xl polynomials.ArticleSYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS. 7:107 (2011)journal articl
Casoratian identities for the Wilson and Askey-Wilson polynomials
No full textInfinitely many Casoratian identities are derived for the Wilson and Askey–Wilson polynomials in parallel to the Wronskian identities for the Hermite, Laguerre and Jacobi polynomials, which were reported recently by the present authors. These identities form the basis of the equivalence between eigenstate adding and deleting Darboux transformations for solvable (discrete) quantum mechanical systems. Similar identities hold for various reduced form polynomials of the Wilson and Askey–Wilson polynomials, e.g. the continuous qq-Jacobi, continuous (dual) (qq-)Hahn, Meixner–Pollaczek, Al-Salam–Chihara, continuous (big) qq-Hermite, etc
Recurrence relations of the multi-indexed orthogonal polynomials. IV. Closure relations and creation/annihilation operators
No full textWe consider the exactly solvable quantum mechanical systems whose eigenfunctions are described by the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types. Corresponding to the recurrence relations with constant coefficients for the M-indexed orthogonal polynomials, it is expected that the systems satisfy the generalized closure relations. In fact we can verify this statement for small M examples. The generalized closure relation gives the exact Heisenberg operator solution of a certain operator, from which the creation and annihilation operators of the system are obtained. Published by AIP Publishing
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