divided differences
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11: 18.30 Associated OP’s
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►The ratio , as defined here, thus provides the same statement of Markov’s Theorem, as in (18.2.9_5), but now in terms of differently obtained numerator and denominator polynomials.
…Ismail (2009, §2.6) discusses this in a different
notation; also note the assumption that , made throughout that reference, Ismail (2009, p. 16).
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12: 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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13: 27.5 Inversion Formulas
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27.5.2
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14: 27.14 Unrestricted Partitions
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►Multiplying the power series for with that for and equating coefficients, we obtain the recursion formula
…Logarithmic differentiation of the generating function leads to another recursion:
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27.14.7
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►where (Hardy and Ramanujan (1918)).
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►where and is given by (27.14.11).
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15: 7.12 Asymptotic Expansions
16: 1.10 Functions of a Complex Variable
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►It should be noted that different branches of used in forming in (1.10.16) give rise to different solutions of (1.10.12).
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17: 18.2 General Orthogonal Polynomials
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►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).
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►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)).
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18: 4.26 Integrals
19: 21.1 Special Notation
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positive integers. | |
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set of all elements of , modulo elements of . Thus two elements of are equivalent if they are both in and their difference is in . (For an example see §20.12(ii).) | |
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