x-difference operators
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31—40 of 86 matching pages
31: 16.21 Differential Equation
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16.21.1
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32: 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.
33: Philip J. Davis
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►At that time John Todd was Chief of the Numerical Analysis Section of the Applied Mathematics Division and head of the Computation Laboratory that co-developed, with the NBS Electronic Computer Laboratory, the Standards Eastern Automatic Computer (SEAC), the first fully operational stored-program electronic digital computer in the United States.
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34: 31.10 Integral Equations and Representations
35: 18.2 General Orthogonal Polynomials
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§18.2(ii) -Difference Operators
►If the orthogonality discrete set is or , then the role of the differentiation operator in the case of classical OP’s (§18.3) is played by , the forward-difference operator, or by , the backward-difference operator; compare §18.1(i). … ►If the orthogonality interval is or , then the role of can be played by , the central-difference operator in the imaginary direction (§18.1(i)). … ►The operator is a delta operator, i. … …36: 2.7 Differential Equations
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2.7.23
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►provided that .
…and denotes the variational operator (§2.3(i)).
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2.7.25
►Assuming also , we have
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37: 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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38: 36.10 Differential Equations
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►In terms of the normal form (36.2.1) the satisfy the operator equation
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§36.10(iii) Operator Equations
►In terms of the normal forms (36.2.2) and (36.2.3), the satisfy the following operator equations …39: Mathematical Introduction
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complex plane (excluding infinity). | |
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(or ) | forward difference operator: . |
(or ) | backward difference operator: . (See also del operator in the Notations section.) |
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