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11: 18.19 Hahn Class: Definitions
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►The Askey scheme extends the three families of classical OP’s (Jacobi, Laguerre and Hermite) with eight further families of OP’s for which the role of the differentiation operator in the case of the classical OP’s is played by a suitable difference operator.
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Hahn class (or linear lattice class). These are OP’s where the role of is played by or or (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.
Wilson class (or quadratic lattice class). These are OP’s ( of degree in , quadratic in ) where the role of the differentiation operator is played by or or . The Wilson class consists of two discrete and two continuous families.
12: 23.21 Physical Applications
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►The Weierstrass function
plays a similar role for cubic potentials in canonical form .
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13: 25.15 Dirichlet -functions
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►This result plays an important role in the proof of Dirichlet’s theorem on primes in arithmetic progressions (§27.11).
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14: 18.25 Wilson Class: Definitions
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►For the Wilson class OP’s with : if the -orthogonality set is , then the role of the differentiation operator in the Jacobi, Laguerre, and Hermite cases is played by the operator followed by division by , or by the operator followed by division by .
Alternatively if the -orthogonality interval is , then the role of is played by the operator followed by division by .
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15: 18.38 Mathematical Applications
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►Classical OP’s play a fundamental role in Gaussian quadrature.
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►Hermite polynomials (and their Freud-weight analogs (§18.32)) play an important role in random matrix theory.
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16: Possible Errors in DLMF
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►Errors in the printed Handbook may already have been corrected in the online version; please consult Errata.
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17: DLMF Project News
error generating summary18: Frank W. J. Olver
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►Olver was an applied mathematician of world renown, one of the most widely recognized contemporary scholars in the field of special functions.
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►was published by World Scientific Publishing Co.
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►Most notably, he served as the Editor-in-Chief and Mathematics Editor of the online NIST Digital Library of Mathematical Functions and its 966-page print companion, the NIST Handbook of Mathematical Functions (Cambridge University Press, 2010).
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19: How to Cite
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►The direct correspondence between the reference numbers in the printed Handbook and the permalinks used online in the DLMF enables readers of either version to cite specific items and their readers to easily look them up again — in either version!
►The following table outlines the correspondence between reference numbers as they appear in the Handbook, and the URL’s that find the same item online.
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