Pochhammer Notations Traditional name Pochhammer symbol Traditional notation HaL n Mathematica StandardForm notation Pochhammer@a, nD Primary definition 06.10.02.0001.01 HaL n GHa + nL GHaL ; HH-a ˛ Z -a ‡ 0 n ˛ Z n £-aLL 06.10.02.0002.01 HaL n k=0 n-1 Ha + kL; n ˛ N + For Α= a, Ν n integers with a £ 0, n £-a, the Pochhammer symbol HΑL Ν can not be uniquely defined by a limiting procedure based on the above definition because the two variables Α, Ν can approach these integers a, n with a £ 0, n £-a at different speeds. For such integers with a £ 0, n £-a we define: 06.10.02.0003.01 HaL n H-1L n H-aL ! H-a - nL ! ; -a ˛ N n ˛ Z n £-a Specific values Specialized values For fixed a 06.10.03.0001.01 HaL n k=0 n-1 Ha + kL; n ˛ N +
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Pochhammer
Notations
Traditional name
Pochhammer symbol
Traditional notation
HaLn
Mathematica StandardForm notation
Pochhammer@a, nD
Primary definition06.10.02.0001.01
HaLn �GHa + nL
GHaL �; HØ H-a Î Z ß -a ³ 0 ß n Î Z ß n £ -aLL06.10.02.0002.01
HaLn � äk=0
n-1 Ha + kL �; n Î N+
For Α = a, Ν � n integers with a £ 0, n £ -a, the Pochhammer symbol HΑLΝ can not be uniquely defined by a limiting
procedure based on the above definition because the two variables Α, Ν can approach these integers a, n with a £ 0, n £ -a
at different speeds. For such integers with a £ 0, n £ -a we define:
06.10.02.0003.01
HaLn �H-1Ln H-aL !
H-a - nL !�; -a Î N ß n Î Z ß n £ -a
Specific values
Specialized values
For fixed a
06.10.03.0001.01
HaLn � äk=0
n-1 Ha + kL �; n Î N+
06.10.03.0002.01HaL0 � 1
06.10.03.0003.01HaL1 � a
06.10.03.0004.01HaL2 � a Ha + 1L06.10.03.0005.01HaL3 � a Ha + 1L Ha + 2L06.10.03.0006.01HaL4 � a Ha + 1L Ha + 2L Ha + 3L06.10.03.0007.01HaL5 � a Ha + 1L Ha + 2L Ha + 3L Ha + 4L06.10.03.0008.01
HaL-n � äk=1
n 1
a - k�; n Î N+
06.10.03.0009.01
HaL-5 �1
Ha - 1L Ha - 2L Ha - 3L Ha - 4L Ha - 5L06.10.03.0010.01
HaL-4 �1
Ha - 1L Ha - 2L Ha - 3L Ha - 4L06.10.03.0011.01
HaL-3 �1
Ha - 1L Ha - 2L Ha - 3L06.10.03.0012.01
HaL-2 �1
Ha - 1L Ha - 2L06.10.03.0013.01
HaL-1 �1
a - 1
For fixed n
06.10.03.0014.01
H0L-n �H-1Ln
n!�; n Î N+
06.10.03.0015.01H0Ln � 0 �; n Î N+
06.10.03.0016.01
-1
2 n� -
H2 n - 2L !
22 n-1 Hn - 1L !
http://functions.wolfram.com 2
06.10.03.0017.01
1
2 n�
H2 n - 1L !
22 n-1 Hn - 1L !
06.10.03.0018.01H1Ln � n!
06.10.03.0019.01
3
2 n�
H2 n + 1L !
4n n!
Values at fixed points
06.10.03.0020.01H0L0 � 1
General characteristics
Domain and analyticity
HaLn is an analytical function of a and n which is defined over C2.
06.10.04.0001.01Ha * nL �HaLn � HC Ä CL �C
Symmetries and periodicities
Mirror symmetry
06.10.04.0002.01HaLn � HaLn
Periodicity
No periodicity
Poles and essential singularities
With respect to n
For fixed a, the function HaLn has an infinite set of singular points:
a) n � -a - k �; k Î N are the simple poles with residues H-1Lk
k! GHaL ;
b) n � ¥� is the point of convergence of poles, which is an essential singular point.
06.10.04.0003.01
SingnIHaLnM � 888-a - k, 1< �; k Î N<, 8¥� , ¥<<06.10.04.0004.01
resnIHaLnM H-a - kL �H-1Lk
k ! GHaL �; k Î N
With respect to a
http://functions.wolfram.com 3
For fixed n, the function HaLn has an infinite set of singular points:
a) a � -k - n �; k Î N, are the simple poles with residues H-1Lk
k! GH-n-kL �; k + n Ï N;
b) a � ¥� is the point of convergence of poles, which is an essential singular point.
06.10.04.0005.01
SingaIHaLnM � 888-k - n, 1< �; k Î N<, 8¥� , ¥<<06.10.04.0006.01
resaIHaLnM H-k - nL �H-1Lk
k ! GH-n - kL �; k Î N ì k + n Ï N
Branch points
With respect to n
The function HaLn does not have branch points with respect to n.
06.10.04.0007.01
BPnIHaLnM � 8<With respect to a
The function HaLn does not have branch points with respect to a.
06.10.04.0008.01
BPaIHaLnM � 8<Branch cuts
With respect to n
The function HaLn does not have branch cuts with respect to n.
06.10.04.0009.01
BCnIHaLnM � 8<With respect to a
The function HaLn does not have branch cuts with respect to a.
06.10.04.0010.01
BCaIHaLnM � 8<Series representations
Generalized power series
Expansions at a � 0
For the function itself
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General case
06.10.06.0008.01
HaLn µ G HnL a + GHnL HΨHnL + ýL a2 + GHnL 1
12I6 ý2 - Π2M + ý ΨHnL +
1
2IΨHnL2 + ΨH1LHnLM a3 + ¼ �; Ha ® 0L
06.10.06.0009.01
HaLn µ G HnL a + GHnL HΨHnL + ýL a2 + GHnL 1
12I6 ý2 - Π2M + ý ΨHnL +
1
2IΨHnL2 + ΨH1LHnLM a3 + OIa4M
06.10.06.0010.01HaLn µ a GHnL + OIa2MSpecial cases
06.10.06.0014.01Ha + ΕLn µ HaLn H1 + OHΕLL �; Ø Ha Î Z ß a £ 0L06.10.06.0015.01Ha + ΕLn µ HaLn I1 + HΨHa + nL - ΨHaLL Ε + OIΕ2MM �; Ø Ha Î Z ß a £ 0L06.10.06.0016.01