Suggested 2013-06-24 by Christopher Gilbreth
The notation of §18.2(iv) will be used.
18.9.1 | |||
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with initial values and .
For ,
18.9.2 | ||||
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and have to be understood for or by continuity in and , that is, and .
Reported 2010-09-16 by Kendall Atkinson
18.9.2_1 | |||
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with initial values and .
For ,
18.9.2_2 | ||||
and have to be understood for or by continuity in and , that is, and and .
For the other classical OP’s see Table 18.9.2.
18.9.3 | |||
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18.9.4 | |||
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18.9.5 | |||
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18.9.6 | |||
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and a similar pair to (18.9.5) and (18.9.6) by symmetry; compare the second row in Table 18.6.1.
18.9.7 | ||||
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18.9.8 | ||||
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18.9.9 | ||||
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18.9.10 | ||||
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18.9.11 | ||||
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18.9.12 | ||||
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Identities similar to (18.9.11) and (18.9.12) involving and can be obtained using rows 4 and 7 in Table 18.6.1.
18.9.13 | ||||
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18.9.14 | ||||
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Formulas (18.9.5), (18.9.11), (18.9.13) are special cases of (18.2.16). Formulas (18.9.6), (18.9.12), (18.9.14) are special cases of (18.2.17).
18.9.15 | |||
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18.9.16 | |||
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Further -th derivative formulas relating two different Jacobi polynomials can be obtained from §15.5(i) by substitution of (18.5.7).
Formula (18.9.15) is degree lowering, while it raises the parameters. Formula (18.9.16) is degree raising, while it lowers the parameters. The following three formulas change the degree but preserve the parameters, see (18.2.42)–(18.2.44) for similar formulas for more general OP’s.
18.9.17 | |||
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18.9.18 | |||
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and the structure relation
18.9.18_5 | |||
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18.9.19 | |||
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18.9.20 | |||
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See also the differentiation formulas in (Erdélyi et al., 1953b, §10.9(15))).
18.9.21 | |||
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18.9.22 | |||
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18.9.23 | ||||
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18.9.24 | ||||
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Further -th derivative formulas relating two different Laguerre polynomials can be obtained from §13.3(ii) by substitution of (13.6.19).
18.9.25 | ||||
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18.9.26 | ||||
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18.9.27 | ||||
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18.9.28 | ||||
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