Pochhammer symbol
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11—20 of 125 matching pages
11: 18.27 -Hahn Class
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18.27.9
, , ,
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- Symbols:
- : multiple -Pochhammer symbol, : real variable, : real variable and
- Permalink:
- http://dlmf.nist.gov/18.27.E9
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.2):
- -Pochhammersymbols now link to their definition.
- See also:
- Annotations for §18.27(iii), §18.27 and Ch.18
18.27.9_5
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- Symbols:
- : binomial coefficient, : -Pochhammer symbol (or -shifted factorial), : multiple -Pochhammer symbol, : real variable, : nonnegative integer and
- Referenced by:
- §18.27(iii), Erratum (V1.2.0) Section 18.27
- Permalink:
- http://dlmf.nist.gov/18.27.E9_5
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.2.0):
- This equation was added.
- See also:
- Annotations for §18.27(iii), §18.27 and Ch.18
18.27.12
, .
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- Symbols:
- : multiple -Pochhammer symbol, : real variable, : real variable and
- Permalink:
- http://dlmf.nist.gov/18.27.E12
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.2):
- -Pochhammersymbols now link to their definition.
- See also:
- Annotations for §18.27(iii), §18.27 and Ch.18
18.27.14_1
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : multiple -Pochhammer symbol, : real variable, : nonnegative integer and
- Source:
- Koekoek et al. (2010, (14.12.2))
- Referenced by:
- §18.27(iv), (18.28.27), Erratum (V1.2.0) Section 18.27
- Permalink:
- http://dlmf.nist.gov/18.27.E14_1
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.2.0):
- This equation was added.
- See also:
- Annotations for §18.27(iv), §18.27 and Ch.18
18.27.19
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- Symbols:
- : Kronecker delta, : Stieltjes–Wigert polynomial, : differential of , : integral, : principal branch of logarithm function, : -Pochhammer symbol (or -shifted factorial), : multiple -Pochhammer symbol, : real variable, : nonnegative integer, : nonnegative integer and : real variable
- Permalink:
- http://dlmf.nist.gov/18.27.E19
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.27(vi), §18.27 and Ch.18
12: 18.29 Asymptotic Approximations for -Hahn and Askey–Wilson Classes
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18.29.1
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- Symbols:
- : base of natural logarithm, : imaginary unit, : multiple -Pochhammer symbol, : real variable and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/18.29.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.29 and Ch.18
18.29.2
; fixed.
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- Symbols:
- : asymptotic equality, : multiple -Pochhammer symbol, : complex variable, : real variable and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/18.29.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.29 and Ch.18
13: 17.14 Constant Term Identities
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17.14.1
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : nonnegative integer, : complex base, : nonnegative integer, : nonnegative integer and : real variable
- Referenced by:
- §17.14
- Permalink:
- http://dlmf.nist.gov/17.14.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.14, §17.14 and Ch.17
17.14.2
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer, : complex variable and : LHS of (17.2.50)
- Referenced by:
- §17.14
- Permalink:
- http://dlmf.nist.gov/17.14.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.14, §17.14 and Ch.17
17.14.3
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer, : complex variable and : LHS of (17.2.49)
- Permalink:
- http://dlmf.nist.gov/17.14.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.14, §17.14 and Ch.17
17.14.4
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer, : complex variable and : LHS of (17.2.49)
- Permalink:
- http://dlmf.nist.gov/17.14.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.14, §17.14 and Ch.17
17.14.5
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer, : complex variable and : LHS of (17.2.50)
- Referenced by:
- §17.14
- Permalink:
- http://dlmf.nist.gov/17.14.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.14, §17.14 and Ch.17
14: 5.18 -Gamma and -Beta Functions
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5.18.1
,
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : real or complex variable, : nonnegative integer, : nonnegative integer and : real or complex variable
- Permalink:
- http://dlmf.nist.gov/5.18.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.18(i), §5.18 and Ch.5
5.18.2
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- Defines:
- : -factorial (as in )
- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : real or complex variable and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/5.18.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.18(i), §5.18 and Ch.5
5.18.3
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : real or complex variable, : nonnegative integer and : real or complex variable
- Permalink:
- http://dlmf.nist.gov/5.18.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.18(i), §5.18 and Ch.5
5.18.4
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- Defines:
- : -gamma function
- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : real or complex variable and : complex variable
- Permalink:
- http://dlmf.nist.gov/5.18.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.18(ii), §5.18 and Ch.5
5.18.12
, , .
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- Symbols:
- : integral, : -beta function, : -Pochhammer symbol (or -shifted factorial), : -differential, : real part, : real or complex variable, : real or complex variable and : real or complex variable
- Permalink:
- http://dlmf.nist.gov/5.18.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §5.18(iii), §5.18 and Ch.5
15: 18.28 Askey–Wilson Class
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18.28.1
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- Defines:
- : Askey–Wilson polynomial
- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : multiple -Pochhammer symbol, : polynomial of degree , : real variable, : nonnegative integer, : nonnegative integer and : real variable
- Referenced by:
- (18.28.29), §18.28(ii), §18.38(iii), Erratum (V1.2.0) for Equation (18.28.1)
- Permalink:
- http://dlmf.nist.gov/18.28.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.28(ii), §18.28 and Ch.18
18.28.3
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : cosine function, : base of natural logarithm, : imaginary unit, : -Pochhammer symbol (or -shifted factorial), : multiple -Pochhammer symbol, : sine function, : weight function and : real variable
- Permalink:
- http://dlmf.nist.gov/18.28.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18
18.28.4
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : multiple -Pochhammer symbol, : real variable and
- Permalink:
- http://dlmf.nist.gov/18.28.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18
18.28.5
.
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : multiple -Pochhammer symbol, : real variable, : nonnegative integer and
- Permalink:
- http://dlmf.nist.gov/18.28.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18
18.28.22
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : multiple -Pochhammer symbol, : arbitrary small positive constant, : real variable, : nonnegative integer and
- Permalink:
- http://dlmf.nist.gov/18.28.E22
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.28(viii), §18.28 and Ch.18
16: 17.6 Function
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17.6.3
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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/17.6.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17
17.6.5
.
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : multiple -Pochhammer symbol and : complex base
- Permalink:
- http://dlmf.nist.gov/17.6.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17
17.6.10
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : multiple -Pochhammer symbol, : complex base, : nonnegative integer and : complex variable
- Permalink:
- http://dlmf.nist.gov/17.6.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17
17.6.14
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : multiple -Pochhammer symbol, : complex base, : nonnegative integer and : complex variable
- Referenced by:
- §17.6(ii)
- Permalink:
- http://dlmf.nist.gov/17.6.E14
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17
17.6.25
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : multiple -Pochhammer symbol, : complex base, : nonnegative integer, : complex variable and : -differential operator
- Permalink:
- http://dlmf.nist.gov/17.6.E25
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.6(iv), §17.6(iv), §17.6 and Ch.17
17: 17.4 Basic Hypergeometric Functions
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17.4.1
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- Defines:
- or : basic hypergeometric (or -hypergeometric) function
- Symbols:
- : binomial coefficient, : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer, : nonnegative integer, : nonnegative integer and : complex variable
- Referenced by:
- §17.4(iv), §17.4(iv), §17.4(iv), §17.4(iv), §17.4(iv), §18.27(i), §18.27(ii)
- Permalink:
- http://dlmf.nist.gov/17.4.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.4(i), §17.4 and Ch.17
17.4.5
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- Defines:
- : first -Appell function
- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer, : nonnegative integer, : real variable and : real variable
- Referenced by:
- Erratum (V1.0.10) for Section 17.1
- Permalink:
- http://dlmf.nist.gov/17.4.E5
- 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 §17.4(iii), §17.4 and Ch.17
17.4.6
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- Defines:
- : second -Appell function
- Symbols:
- : -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 (17.4.6)
- Permalink:
- http://dlmf.nist.gov/17.4.E6
- Encodings:
- pMML, png, TeX
- Clarification (effective with 1.1.3):
- The explicit usage of in the denominator of the right-hand side was used.
- 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 §17.4(iii), §17.4 and Ch.17
17.4.7
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- Defines:
- : third -Appell function
- Symbols:
- : -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
- Permalink:
- http://dlmf.nist.gov/17.4.E7
- 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 §17.4(iii), §17.4 and Ch.17
17.4.8
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- Defines:
- : fourth -Appell function
- Symbols:
- : -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
- Permalink:
- http://dlmf.nist.gov/17.4.E8
- 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 §17.4(iii), §17.4 and Ch.17
18: 17.7 Special Cases of Higher Functions
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17.7.1
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : multiple -Pochhammer symbol and : complex base
- Referenced by:
- Erratum (V1.1.10) for Equation (17.7.1)
- Permalink:
- http://dlmf.nist.gov/17.7.E1
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.10):
- The constraint is not necessary and was removed.
- See also:
- Annotations for §17.7(i), §17.7(i), §17.7 and Ch.17
17.7.3
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : multiple -Pochhammer symbol and : complex base
- Referenced by:
- §17.6(i), §17.7(i)
- Permalink:
- http://dlmf.nist.gov/17.7.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.7(i), §17.7(i), §17.7 and Ch.17
17.7.18
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), or : basic hypergeometric (or -hypergeometric) function, : multiple -Pochhammer symbol, : complex base, : nonnegative integer and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/17.7.E18
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17
17.7.20
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- Symbols:
- : -Pochhammer symbol (or -shifted factorial), : nonnegative integer, : complex base and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/17.7.E20
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17
17.7.23
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- Symbols:
- : Kronecker delta, : binomial coefficient, : -Pochhammer symbol (or -shifted factorial), : multiple -Pochhammer symbol, : nonnegative integer, : complex base and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/17.7.E23
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17
19: 10.31 Power Series
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10.31.3
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- Symbols:
- : gamma function, : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : modified Bessel function of the first kind, : nonnegative integer, : complex variable and : complex parameter
- Referenced by:
- §10.31
- Permalink:
- http://dlmf.nist.gov/10.31.E3
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.1.2):
- The Pochhammersymbol now links to its definition.
- See also:
- Annotations for §10.31 and Ch.10
20: 18.1 Notation
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-Pochhammer Symbol
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18.1.1
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- Symbols:
- : Chebyshev polynomial of the first kind, : Jacobi polynomial, : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : ultraspherical (or Gegenbauer) polynomial, : nonnegative integer and : real variable
- Permalink:
- http://dlmf.nist.gov/18.1.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.1(iii), §18.1 and Ch.18
18.1.2
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- Defines:
- : shifted Jacobi polynomial
- Symbols:
- : Jacobi polynomial, : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : real variable, : nonnegative integer and : real variable
- Permalink:
- http://dlmf.nist.gov/18.1.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §18.1(iii), §18.1 and Ch.18