Bailey 2F1(-1) sum
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1: 17.12 Bailey Pairs
§17.12 Bailey Pairs
►Bailey Transform
… ►Bailey Pairs
… ►Weak Bailey Lemma
… ►Strong Bailey Lemma
…2: 32.10 Special Function Solutions
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►For example, if , with , then the Riccati equation is
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►with , and , , independently.
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►with and .
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►where , , and , with , , independently.
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►where , , , , and , with , , independently.
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3: 4.24 Inverse Trigonometric Functions: Further Properties
4: 4.38 Inverse Hyperbolic Functions: Further Properties
5: 4.13 Lambert -Function
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►The decreasing solution can be identified as .
…where .
is a single-valued analytic function on , real-valued when , and has a square root branch point at .
…The other branches are single-valued analytic functions on , have a logarithmic branch point at , and, in the case , have a square root branch point at respectively.
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►where for , for on the relevant branch cuts,
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6: 5.5 Functional Relations
7: 32.8 Rational Solutions
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►Then for
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►Then has a rational solution iff one of the following holds with and :
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(c)
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(d)
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►where , , , , and , with , , independently, and at least one of , , or is an integer.
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, , and , with even.
, , and , with even.
8: 10.38 Derivatives with Respect to Order
9: 14.28 Sums
§14.28 Sums
… ►where the branches of the square roots have their principal values when and are continuous when . … ►where and are ellipses with foci at , being properly interior to . The series converges uniformly for outside or on , and within or on . … ►§14.28(iii) Other Sums
…10: 18.18 Sums
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►Let be analytic within an ellipse with foci , and
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►See §3.11(ii), or set in the above results for Jacobi and refer to (18.7.3)–(18.7.6).
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►In all three cases of Jacobi, Laguerre and Hermite, if is on the corresponding interval with respect to the corresponding weight function and if are given by (18.18.1), (18.18.5), (18.18.7), respectively, then the respective series expansions (18.18.2), (18.18.4), (18.18.6) are valid with the sums converging in sense.
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►For the Poisson kernel of Jacobi polynomials (the Bailey formula) see Bailey (1938).
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