Ramanujan Theta Functions
Ramanujan's two-variable theta function f(a,b) is defined by
for |ab|<1 (Berndt 1985, p. 34; Berndt et al. 2000). It satisfies
| f(-1,a)=0 |
(2)
|
and
(Berndt 1985, pp. 34-35; Berndt et al. 2000), where (a;q)_k is a q-Pochhammer symbol, i.e., a q-series.
A one-argument form of f(a,b) is also defined by
(OEIS A010815; Berndt 1985, pp. 36-37; Berndt et al. 2000), where (a;q)_infty is a q-Pochhammer symbol. The identities above are equivalent to the pentagonal number theorem.
The function also satisfies
Ramanujan's phi-function phi(q) is defined by
(OEIS A000122), where theta_3(0,q) is a Jacobi theta function (Berndt 1985, pp. 36-37). f(a,b) is a generalization of phi(x), with the two being connected by
| f(x,x)=phi(x). |
(15)
|
Special values of phi include
where Gamma(x) is a gamma function.
Ramanujan's psi-function psi(q) is defined by
(OEIS A010054; Berndt 1985, p. 37).
Ramanujan's chi-function chi(q) is defined by
(OEIS A000700; Berndt 1985, p. 37).
A different phi function is sometimes defined as
where theta_i(0,q) is again a Jacobi theta function, which has special value
| phi^'(-e^(-pisqrt(3)))=(4sqrt(3)-7)^(1/8). |
(28)
|
See also
Jacobi Triple Product, Pentagonal Number Theorem, q-Series, q-Series Identities, Ramanujan phi-Function, Rogers-Ramanujan Continued Fraction, Schröter's FormulaExplore with Wolfram|Alpha
More things to try:
References
Berndt, B. C. Ramanujan's Notebooks, Part III. New York: Springer-Verlag, 1985.Berndt, B. C.; Huang, S.-S.; Sohn, J.; and Son, S. H. "Some Theorems on the Rogers-Ramanujan Continued Fraction in Ramanujan's Lost Notebook." Trans. Amer. Math. Soc. 352, 2157-2177, 2000.Mc Laughlin, J.; Sills, A. V.; and Zimmer, P. "Dynamic Survey DS15: Rogers-Ramanujan-Slater Type Identities." Electronic J. Combinatorics, DS15, 1-59, May 31, 2008. http://www.combinatorics.org/Surveys/ds15.pdf.Sloane, N. J. A. Sequences A000122, A000700/M0217, A010054, and A010815 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Ramanujan Theta FunctionsCite this as:
Weisstein, Eric W. "Ramanujan Theta Functions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RamanujanThetaFunctions.html