We show that certain two-term transformation formulas between basic hypergeometric series can easily be described by means of invariance groups. For the transformations of nonterminating φ23 series, and those of terminating balanced φ34 series, these invariance groups are symmetric groups. For transformations of φ12 series the invariance group is the dihedral group of order 12. Transformations of terminating φ23 series are described by means of some subgroup of S6, and finally the invariance group of transformations of very-well-poised nonterminating φ78 series is shown to be isomorphic to the Weyl group of a root system of type D5.

1.
G. Gasper and M. Rahman, Basic Hypergeometric Series (Cambridge U.P., Cambridge, 1990).
2.
C.
Krattenthaler
, “
HYP and HYPQ: Mathematica packages for the manipulation of binomial sums and hypergeometric series, respectively q-binomial sums and basic hypergeometric series
,”
J. Symbolic Comput.
20
,
737
–
744
(
1995
). Also available on http://radon.mat.univie.ac.at/People/kratt.
Google Scholar
Crossref
Search ADS
 
3.
F. J. W.
Whipple
, “
A group of generalized hypergeometric series: relations between 120 allied series of the type F[a,b,c;d,e],
”
Proc. London Math. Soc. (2)
23
,
104
–
114
(
1925
).
Google Scholar
 
4.
W. A.
Beyer
,
J. D.
Louck
, and
P. R.
Stein
, “
Group theoretical basis of some identities for the generalized hypergeometric series
,”
J. Math. Phys.
28
,
497
–
508
(
1987
).
Google Scholar
Crossref
Search ADS
 
5.
K. Srinivasa
Rao
,
J.
Van der Jeugt
,
J.
Raynal
,
R.
Jagannathan
, and
V.
Rajeswari
, “
Group theoretical basis for the terminating F23(1) series
,”
J. Phys. A
25
,
861
–
876
(
1992
).
Google Scholar
Crossref
Search ADS
 
6.
J. D. Louck, W. A. Beyer, L. C. Biedenharn, and P. R. Stein, “Symmetries of some hypergeometric series: implications for 3j- and 6j-coefficients,” in XV International Colloquium on Group Theoretical Methods in Physics, edited by R. Gilmore (World Scientific, Singapore, 1987).
7.
For an overview of the relations between 3-j and 6-j coefficients and hypergeometric series, see L. C. Biedenharn and J. D. Louck, The Racah-Wigner Algebra in Quantum Theory. Encyclopedia of Math. and its Applications, Vol. 9. (Addison–Wesley, Reading, MA, 1981), Topic 11, p. 429.
8.
V.
Rajeswari
and
K. Srinivasa
Rao
, “
Four sets of F23(1) functions, Hahn polynomials and recurrence relations for the 3-j coefficients
,”
J. Phys. A
22
,
4113
–
4123
(
1989
).
Google Scholar
Crossref
Search ADS
 
9.
V.
Rajeswari
and
K. Srinivasa
Rao
, “
Generalized basic hypergeometric functions and the q-analogues of 3-j and 6-j coefficients
,”
J. Phys. A
24
,
3761
–
3780
(
1991
).
Google Scholar
Crossref
Search ADS
 
10.
V.
Rajeswari
and
K. Srinivasa
Rao
, “
Note on the Explicit Forms for the Clebsch-Gordan Coefficient of the Quantum Group SUq(2),
”
J. Phys. Soc. Jpn.
60
,
3583
–
3584
(
1991
).
Google Scholar
Crossref
Search ADS
 
11.
M. Hamermesh, Group Theory, and its Application to Physical Problems (Addison–Wesley, Reading, MA, 1962).
12.
J. Thomae, “Über die Funktionen welche durch Reihen von der Form dargestellt werden: 1+pp′p″/1q′q″+⋯ ,” J. Reine Angew. Math. 87 26–73 (1879).
13.
W. N. Bailey, Generalized Hypergeometric Series (Cambridge U.P., Cambridge, 1935).
14.
G. H. Hardy, Ramanujan. Twelve lectures on subjects suggested by his life and work (Cambridge U.P., Cambridge, 1940).
15.
D. B.
Sears
, “
On the transformation theory of basic hypergeometric functions
,”
Proc. London Math. Soc. (2)
53
,
158
–
180
(
1951
).
Google Scholar
 
16.
G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and its Applications, Vol. 16 (Addison–Wesley, Reading, MA 1981).
17.
E. Heine, Handbuch der Kugelfunctionen. Theorie und Anwendungen, (Reimer, Berlin, 1878), Vol. 1.
18.
F. J. Budden, The Fascination of Groups (Cambridge U.P., Cambridge, 1972).
19.
W. N.
Bailey
, “
An identity involving Heine’s basic hypergeometric series
,”
J. London Math. Soc.
4
,
254
–
257
(
1929
).
Google Scholar
 
20.
J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Vol. 9 (Springer, Berlin, 1978).
This content is only available via PDF.
© 1999 American Institute of Physics.
1999
American Institute of Physics
You do not currently have access to this content.