Pochhammer symbol
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1—10 of 125 matching pages
1: 17.2 Calculus
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17.2.6
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : multiple -Pochhammer symbol, : complex base, : nonnegative integer and : nonnegative integer
- Referenced by:
- §17.2(i)
- Permalink:
- http://dlmf.nist.gov/17.2.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.2(i), §17.2 and Ch.17
17.2.19
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : multiple -Pochhammer symbol, : complex base and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/17.2.E19
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.2(i), §17.2 and Ch.17
17.2.20
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : multiple -Pochhammer symbol, : nonnegative integer, : complex base and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/17.2.E20
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.2(i), §17.2 and Ch.17
2: 5.2 Definitions
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§5.2(iii) Pochhammer’s Symbol
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5.2.5
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- Symbols:
- : gamma function, : Pochhammer’s symbol (or shifted factorial), : nonnegative integer and : real or complex variable
- A&S Ref:
- 6.1.22
- Referenced by:
- (25.11.28), (25.5.7), (25.8.2)
- Permalink:
- http://dlmf.nist.gov/5.2.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.2(iii), §5.2 and Ch.5
5.2.6
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- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : nonnegative integer and : real or complex variable
- Referenced by:
- §5.2(iii), Erratum (V1.0.17) for Subsection 5.2(iii)
- Permalink:
- http://dlmf.nist.gov/5.2.E6
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.17):
- This equation was added.
- See also:
- Annotations for §5.2(iii), §5.2 and Ch.5
5.2.7
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►Pochhammer symbols (rising factorials) and falling factorials can be expressed in terms of each other via
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- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : nonnegative integer and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/5.2.E7
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.17):
- This equation was added.
- See also:
- Annotations for §5.2(iii), §5.2 and Ch.5
3: 15.15 Sums
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15.15.1
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- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), or : Olver’s hypergeometric function, : nonnegative integer, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
- Referenced by:
- §15.19(i)
- Permalink:
- http://dlmf.nist.gov/15.15.E1
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.2):
- The Pochhammersymbol now links to its definition.
- See also:
- Annotations for §15.15 and Ch.15
4: 17.11 Transformations of -Appell Functions
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17.11.1
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- Symbols:
- : first -Appell function, or : basic hypergeometric (or -hypergeometric) function, : multiple -Pochhammer symbol, : complex base, : real variable and : real variable
- Referenced by:
- §17.11, Erratum (V1.0.10) for Section 17.1
- Permalink:
- http://dlmf.nist.gov/17.11.E1
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.0.10):
- The notation for has been updated to that of Gasper and Rahman (2004) to explicitly include the argument.
- See also:
- Annotations for §17.11 and Ch.17
17.11.2
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ⓘ
- Symbols:
- : Pochhammer’s symbol (or shifted factorial), : second -Appell function, : -Pochhammer symbol (or -shifted factorial), : multiple -Pochhammer symbol, : complex base, : nonnegative integer, : nonnegative integer, : real variable and : real variable
- Referenced by:
- Erratum (V1.0.10) for Section 17.1, Erratum (V1.1.3) for Equation (17.11.2)
- Permalink:
- http://dlmf.nist.gov/17.11.E2
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.3):
- The factor originally used in the denominator has been corrected to be .
- Correction (effective with 1.0.10):
- The notation for has been updated to that of Gasper and Rahman (2004) to explicitly include the argument.
- See also:
- Annotations for §17.11 and Ch.17
17.11.3
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- Symbols:
- : third -Appell function, : -Pochhammer symbol (or -shifted factorial), : multiple -Pochhammer symbol, : complex base, : nonnegative integer, : nonnegative integer, : real variable and : real variable
- Referenced by:
- §17.11, Erratum (V1.0.10) for Section 17.1
- Permalink:
- http://dlmf.nist.gov/17.11.E3
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.0.10):
- The notation for has been updated to that of Gasper and Rahman (2004) to explicitly include the argument.
- See also:
- Annotations for §17.11 and Ch.17
17.11.4
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : multiple -Pochhammer symbol, : complex base, : nonnegative integer, : nonnegative integer and : real variable
- Permalink:
- http://dlmf.nist.gov/17.11.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.11 and Ch.17
5: 17.12 Bailey Pairs
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17.12.3
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer, : nonnegative integer, : part of Bailey pair and : part of Bailey pair
- Permalink:
- http://dlmf.nist.gov/17.12.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.12, §17.12 and Ch.17
17.12.4
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer, : part of Bailey pair and : part of Bailey pair
- Permalink:
- http://dlmf.nist.gov/17.12.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.12, §17.12 and Ch.17
6: 17.5 Functions
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17.5.1
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- Symbols:
- : binomial coefficient, : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : complex base, : nonnegative integer and : complex variable
- Referenced by:
- §17.2(iii), Erratum (V1.1.11) for Equation (17.5.1)
- Permalink:
- http://dlmf.nist.gov/17.5.E1
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.11):
- The constraint is not necessary and was removed.
- See also:
- Annotations for §17.5, §17.5 and Ch.17
17.5.2
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : complex base and : complex variable
- Permalink:
- http://dlmf.nist.gov/17.5.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.5, §17.5 and Ch.17
17.5.3
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : complex base, : nonnegative integer and : complex variable
- Permalink:
- http://dlmf.nist.gov/17.5.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.5, §17.5 and Ch.17
17.5.4
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ⓘ
- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : complex base, : nonnegative integer and : complex variable
- Referenced by:
- §17.2(iii)
- Permalink:
- http://dlmf.nist.gov/17.5.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.5, §17.5 and Ch.17
17.5.5
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function and : complex base
- Referenced by:
- Erratum (V1.1.10) for Equation (17.5.5)
- Permalink:
- http://dlmf.nist.gov/17.5.E5
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.10):
- The constraint is not necessary and was removed.
- See also:
- Annotations for §17.5, §17.5 and Ch.17
7: 17.3 -Elementary and -Special Functions
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17.3.1
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- Defines:
- : -exponential function
- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer and : real variable
- Referenced by:
- §17.5
- Permalink:
- http://dlmf.nist.gov/17.3.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17
17.3.2
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- Defines:
- : -exponential function
- Symbols:
- : binomial coefficient, : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer and : real variable
- Referenced by:
- §17.5
- Permalink:
- http://dlmf.nist.gov/17.3.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17
17.3.3
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- Defines:
- : -sine function
- Symbols:
- : imaginary unit, : -Pochhammer symbol (or -shifted factorial), : -exponential function, : complex base, : nonnegative integer and : real variable
- Permalink:
- http://dlmf.nist.gov/17.3.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17
17.3.5
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- Defines:
- : -cosine function
- Symbols:
- : imaginary unit, : -Pochhammer symbol (or -shifted factorial), : -exponential function, : complex base, : nonnegative integer and : real variable
- Permalink:
- http://dlmf.nist.gov/17.3.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17
17.3.6
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- Defines:
- : -cosine function
- Symbols:
- : imaginary unit, : -exponential function, : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer and : real variable
- Permalink:
- http://dlmf.nist.gov/17.3.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17
8: 17.8 Special Cases of Functions
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17.8.1
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- Symbols:
- : multiple -Pochhammer symbol, : complex base, : nonnegative integer and : complex variable
- Permalink:
- http://dlmf.nist.gov/17.8.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.8, §17.8 and Ch.17
17.8.2
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- Symbols:
- or : bilateral basic hypergeometric (or bilateral -hypergeometric) function, : multiple -Pochhammer symbol, : complex base and : complex variable
- Permalink:
- http://dlmf.nist.gov/17.8.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.8, §17.8 and Ch.17
17.8.3
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- Symbols:
- : multiple -Pochhammer symbol, : complex base, : nonnegative integer and : complex variable
- Permalink:
- http://dlmf.nist.gov/17.8.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.8, §17.8 and Ch.17
17.8.4
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ⓘ
- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : bilateral basic hypergeometric (or bilateral -hypergeometric) function, : multiple -Pochhammer symbol and : complex base
- Permalink:
- http://dlmf.nist.gov/17.8.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.8, §17.8 and Ch.17
17.8.8
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : bilateral basic hypergeometric (or bilateral -hypergeometric) function, : multiple -Pochhammer symbol and : complex base
- Source:
- Verma and Jain (1983, (3.9))
- Referenced by:
- §17.6(i), §17.8, Erratum (V1.1.0) for Additions
- Permalink:
- http://dlmf.nist.gov/17.8.E8
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.1.0):
-
This equation was added.
Suggested 2019-10-19 by Slobodan Damjanovic
- See also:
- Annotations for §17.8, §17.8 and Ch.17
9: 17.13 Integrals
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17.13.1
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- Symbols:
- : integral, : -Pochhammer symbol (or -shifted factorial), : -differential, : complex base and : real variable
- Permalink:
- http://dlmf.nist.gov/17.13.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.13 and Ch.17
17.13.2
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- Symbols:
- : integral, : -gamma function, : -Pochhammer symbol (or -shifted factorial), : -differential, : complex base and : real variable
- Permalink:
- http://dlmf.nist.gov/17.13.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.13 and Ch.17
17.13.3
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ⓘ
- Symbols:
- : gamma function, : differential of , : integral, : -gamma function, : -Pochhammer symbol (or -shifted factorial) and : complex base
- Referenced by:
- Erratum (V1.0.1) for Equation (17.13.3)
- Permalink:
- http://dlmf.nist.gov/17.13.E3
- Encodings:
- pMML, png, TeX
- Errata (effective with 1.0.1):
- Originally the differential was identified incorrectly as a -differential; the correct differential is .
- See also:
- Annotations for §17.13, §17.13 and Ch.17
17.13.4
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- Symbols:
- : integral, : -gamma function, : -Pochhammer symbol (or -shifted factorial), : -differential and : complex base
- Permalink:
- http://dlmf.nist.gov/17.13.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.13, §17.13 and Ch.17
10: 17.9 Further Transformations of Functions
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17.9.1
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : complex base and : complex variable
- Permalink:
- http://dlmf.nist.gov/17.9.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17
17.9.3
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ⓘ
- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : multiple -Pochhammer symbol, : complex base and : complex variable
- Referenced by:
- Erratum (V1.0.23) for Equation (17.9.3)
- Permalink:
- http://dlmf.nist.gov/17.9.E3
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.0.23):
-
The second term on the right-hand side was missing.
The form of the equation where the second term is missing is
correct if the is terminating. It is this
form which appeared in the first edition of Gasper and Rahman (1990).
The more general version which appears now is what is reproduced
in Gasper and Rahman (2004, (III.5)).
Suggested 2019-04-26 by Roberto S. Costas-Santos
- See also:
- Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17
17.9.4
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : complex base, : nonnegative integer and : complex variable
- Permalink:
- http://dlmf.nist.gov/17.9.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17
17.9.5
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : complex base, : nonnegative integer and : complex variable
- Permalink:
- http://dlmf.nist.gov/17.9.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17
17.9.8
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : complex base and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/17.9.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17