§14.7 Integer Degree and Order
Contents
- §14.7(i)
- §14.7(ii) Rodrigues-Type Formulas
- §14.7(iii) Reflection Formulas
- §14.7(iv) Generating Functions
§14.7(i)
For ,
14.7.1 |
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where is the Legendre polynomial of degree . For
additional properties of see Chapter 18.
14.7.2 |
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where , and for ,
14.7.3 |
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equivalently,
14.7.4 |
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14.7.5 |
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Next,
14.7.6 |
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where
14.7.7 |
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§14.7(ii) Rodrigues-Type Formulas
For , and ,
14.7.8 |
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14.7.9 |
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14.7.10 |
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14.7.11 |
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14.7.12 |
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14.7.13 |
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14.7.14 |
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14.7.15 |
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When is even and , and
are polynomials of degree . Also,
14.7.16 |
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§14.7(iii) Reflection Formulas
14.7.17 |
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14.7.18 |
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§14.7(iv) Generating Functions
When and ,
14.7.19 |
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14.7.20 |
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When and ,
14.7.21 |
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When , (14.7.19) applies with
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14.7.22 |
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Lastly, when , (14.7.21) applies with
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For other generating functions see Magnus et al. (1966, pp. 232–233) and
Rainville (1960, pp. 163–165, 168, 170–171, 184).
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