alternative forms
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1: 32.2 Differential Equations
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§32.2(iii) Alternative Forms
…2: 22.8 Addition Theorems
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§22.8(ii) Alternative Forms for Sum of Two Arguments
…3: 1.8 Fourier Series
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Alternative Form
…4: 33.2 Definitions and Basic Properties
5: 18.39 Applications in the Physical Sciences
6: Mathematical Introduction
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►This section may also include important alternative notations that have appeared in the literature.
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►Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions (§14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19).
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►In the corresponding section for the DLMF some of the alternative notations that appear in the first section of the special function chapters are also included.
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►For equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55 equation number, or other form of reference, together with corrections, if needed.
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7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►These are based on the Liouville normal form of (1.13.29).
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►Consider formally self-adjoint operators of the form
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Self-adjoint extensions of (1.18.28) and the Weyl alternative
… ►By Weyl’s alternative equals either 1 (the limit point case) or 2 (the limit circle case), and similarly for . …8: 18.38 Mathematical Applications
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18.38.3
, , ,
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►The symbol (34.2.6), with an alternative expression as a terminating of unit argument, can be expressed in terms of Hahn polynomials (18.20.5) or, by (18.21.1), dual Hahn polynomials.
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►The symbol (34.4.3), with an alternative expression as a terminating balanced of unit argument, can be expressend in terms of Racah polynomials (18.26.3).
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►A symmetric Laurent polynomial is a function of the form
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►Exceptional OP’s (EOP’s) are those which are ‘missing’ a finite number of lower order polynomials, but yet form complete sets with respect to suitable measures.
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