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DLMF: Β§17.2 Calculus β£ Properties β£ Chapter 17 π-Hypergeometric and Related Functions
Β§17.2 Calculus
Contents
- Β§17.2(i) -Calculus
- Β§17.2(ii) Binomial Coefficients
- Β§17.2(iii) Binomial Theorem
- Β§17.2(iv) Derivatives
- Β§17.2(v) Integrals
- Β§17.2(vi) RogersβRamanujan Identities
Β§17.2(i) -Calculus
For ,
17.2.1 |
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17.2.2 |
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For
17.2.3 |
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when this product converges.
17.2.4 |
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17.2.5 |
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17.2.6 |
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For properties of the function
see Β§27.14.
Let and . Then |
17.2.6_1 |
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17.2.6_2 |
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For these and similar results see (Apostol, 1990, Ch. 3) and (Katsurada, 2003, Β§3).
Note that (17.2.6_1) is just (27.14.14) with and . |
17.2.7 |
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17.2.8 |
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17.2.9 |
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17.2.10 |
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17.2.11 |
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17.2.12 |
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17.2.13 |
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17.2.14 |
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17.2.15 |
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17.2.16 |
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17.2.17 |
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17.2.18 |
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17.2.19 |
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more generally,
17.2.20 |
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17.2.21 |
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17.2.22 |
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more generally,
17.2.23 |
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where .
17.2.24 |
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17.2.25 |
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17.2.26 |
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Β§17.2(ii) Binomial Coefficients
17.2.27 |
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17.2.28 |
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17.2.29 |
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17.2.30 |
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17.2.31 |
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17.2.32 |
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17.2.33 |
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17.2.34 |
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provided that .
Β§17.2(iii) Binomial Theorem
17.2.35 |
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In the limit as , (17.2.35) reduces to the standard binomial
theorem
17.2.36 |
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Also,
17.2.37 |
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provided that . When , where is a nonnegative
integer, (17.2.37) reduces to the -binomial series
17.2.38 |
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17.2.39 |
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17.2.40 |
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When in (17.2.35), and when in
(17.2.38), the results become convergent infinite series and infinite
products (see (17.5.1) and (17.5.4)).
Β§17.2(iv) Derivatives
The -derivatives of are defined by
17.2.41 |
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and
17.2.42 |
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When the -derivatives converge to the corresponding ordinary
derivatives.
Product Rule
17.2.43 |
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Leibniz Rule
17.2.44 |
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-differential equations are considered in Β§17.6(iv).
Β§17.2(v) Integrals
If is continuous at , then
17.2.45 |
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and more generally,
17.2.46 |
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If is continuous on , then
17.2.47 |
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Infinite Range
17.2.48 |
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provided that converges.
Β§17.2(vi) RogersβRamanujan Identities
17.2.49 |
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17.2.50 |
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These identities are the first in a large collection of similar results. See
Β§17.14.