xdifference operators
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1: 25.17 Physical Applications
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►This relates to a suggestion of Hilbert and Pólya that the zeros are eigenvalues of some operator, and the Riemann hypothesis is true if that operator is Hermitian.
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2: 18.42 Software
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►Also included is a website (CAOP) operated by a university department.
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3: 26.19 Mathematical Applications
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►These have applications in operations research, probability theory, and statistics.
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4: Brian R. Judd
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►Judd’s books include Operator Techniques in Atomic Spectroscopy, published by McGrawHill in 1963 and reprinted by Princeton University Press in 1998, Second
Quantization and Atomic Spectroscopy, published by Johns Hopkins in 1967, Topics in Atomic and Nuclear Theory (with J.
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5: 13.27 Mathematical Applications
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►Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function.
The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions.
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6: 26.22 Software
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►Also included are websites operated by university departments and consortia, research institutions, and peerreviewed journals.
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7: 34.9 Graphical Method
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►The graphical method establishes a onetoone correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression.
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8: Guide to Searching the DLMF
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proximity operator:
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and and or
adj, prec/n, and near/n, where n is any positive natural number.
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9: 18.2 General Orthogonal Polynomials
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§18.2(ii) $x$Difference Operators
►If the orthogonality discrete set $X$ is $\{0,1,\mathrm{\dots},N\}$ or $\{0,1,2,\mathrm{\dots}\}$, then the role of the differentiation operator $d/dx$ in the case of classical OP’s (§18.3) is played by ${\mathrm{\Delta}}_{x}$, the forwarddifference operator, or by ${\nabla}_{x}$, the backwarddifference operator; compare §18.1(i). … ►If the orthogonality interval is $(\mathrm{\infty},\mathrm{\infty})$ or $(0,\mathrm{\infty})$, then the role of $d/dx$ can be played by ${\delta}_{x}$, the centraldifference operator in the imaginary direction (§18.1(i)). …10: About MathML
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