From Wikipedia, the free encyclopedia
In
mathematics , the elliptic gamma function is a generalization of the
q-gamma function , which is itself the
q-analog of the ordinary
gamma function . It is closely related to a function studied by
Jackson (1905) , and can be expressed in terms of the
triple gamma function . It is given by
Γ
(
z
;
p
,
q
)
=
∏
m
=
0
∞
∏
n
=
0
∞
1
−
p
m
+
1
q
n
+
1
/
z
1
−
p
m
q
n
z
.
{\displaystyle \Gamma (z;p,q)=\prod _{m=0}^{\infty }\prod _{n=0}^{\infty }{\frac {1-p^{m+1}q^{n+1}/z}{1-p^{m}q^{n}z}}.}
It obeys several identities:
Γ
(
z
;
p
,
q
)
=
1
Γ
(
p
q
/
z
;
p
,
q
)
{\displaystyle \Gamma (z;p,q)={\frac {1}{\Gamma (pq/z;p,q)}}\,}
Γ
(
p
z
;
p
,
q
)
=
θ
(
z
;
q
)
Γ
(
z
;
p
,
q
)
{\displaystyle \Gamma (pz;p,q)=\theta (z;q)\Gamma (z;p,q)\,}
and
Γ
(
q
z
;
p
,
q
)
=
θ
(
z
;
p
)
Γ
(
z
;
p
,
q
)
{\displaystyle \Gamma (qz;p,q)=\theta (z;p)\Gamma (z;p,q)\,}
where θ is the
q-theta function .
When
p
=
0
{\displaystyle p=0}
, it essentially reduces to the infinite
q-Pochhammer symbol :
Γ
(
z
;
0
,
q
)
=
1
(
z
;
q
)
∞
.
{\displaystyle \Gamma (z;0,q)={\frac {1}{(z;q)_{\infty }}}.}
Multiplication Formula
Define
Γ
~
(
z
;
p
,
q
)
:=
(
q
;
q
)
∞
(
p
;
p
)
∞
(
θ
(
q
;
p
)
)
1
−
z
∏
m
=
0
∞
∏
n
=
0
∞
1
−
p
m
+
1
q
n
+
1
−
z
1
−
p
m
q
n
+
z
.
{\displaystyle {\tilde {\Gamma }}(z;p,q):={\frac {(q;q)_{\infty }}{(p;p)_{\infty }}}(\theta (q;p))^{1-z}\prod _{m=0}^{\infty }\prod _{n=0}^{\infty }{\frac {1-p^{m+1}q^{n+1-z}}{1-p^{m}q^{n+z}}}.}
Then the following formula holds with
r
=
q
n
{\displaystyle r=q^{n}}
(
Felder & Varchenko (2002) ).
Γ
~
(
n
z
;
p
,
q
)
Γ
~
(
1
/
n
;
p
,
r
)
Γ
~
(
2
/
n
;
p
,
r
)
⋯
Γ
~
(
(
n
−
1
)
/
n
;
p
,
r
)
=
(
θ
(
r
;
p
)
θ
(
q
;
p
)
)
n
z
−
1
Γ
~
(
z
;
p
,
r
)
Γ
~
(
z
+
1
/
n
;
p
,
r
)
⋯
Γ
~
(
z
+
(
n
−
1
)
/
n
;
p
,
r
)
.
{\displaystyle {\tilde {\Gamma }}(nz;p,q){\tilde {\Gamma }}(1/n;p,r){\tilde {\Gamma }}(2/n;p,r)\cdots {\tilde {\Gamma }}((n-1)/n;p,r)=\left({\frac {\theta (r;p)}{\theta (q;p)}}\right)^{nz-1}{\tilde {\Gamma }}(z;p,r){\tilde {\Gamma }}(z+1/n;p,r)\cdots {\tilde {\Gamma }}(z+(n-1)/n;p,r).}
References
Felder, G.; Varchenko, A. (2002). "Multiplication Formulas for the Elliptic Gamma Function".
arXiv :
math/0212155 .
Jackson, F. H. (1905), "The Basic Gamma-Function and the Elliptic Functions", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character , 76 (508), The Royal Society: 127–144,
Bibcode :
1905RSPSA..76..127J ,
doi :
10.1098/rspa.1905.0011 ,
ISSN
0950-1207 ,
JSTOR
92601
Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series , Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.),
Cambridge University Press ,
ISBN
978-0-521-83357-8 ,
MR
2128719
Ruijsenaars, S. N. M. (1997),
"First order analytic difference equations and integrable quantum systems" ,
Journal of Mathematical Physics , 38 (2): 1069–1146,
Bibcode :
1997JMP....38.1069R ,
doi :
10.1063/1.531809 ,
ISSN
0022-2488 ,
MR
1434226
Felder, Giovanni; Henriques, André; Rossi, Carlo A.; Zhu, Chenchang (2008). "A gerbe for the elliptic gamma function". Duke Mathematical Journal . 141 .
arXiv :
math/0601337 .
doi :
10.1215/S0012-7094-08-14111-0 .
S2CID
817920 .