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Ramanujan Theta Functions


Ramanujan's two-variable theta function f(a,b) is defined by

 f(a,b)=sum_(n=-infty)^inftya^(n(n+1)/2)b^(n(n-1)/2)
(1)

for |ab|<1 (Berndt 1985, p. 34; Berndt et al. 2000). It satisfies

 f(-1,a)=0
(2)

and

f(a,b)=f(b,a)
(3)
=(-a;ab)_infty(-b;ab)_infty(ab;ab)_infty
(4)

(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

f(-q)=f(-q,-q^2)
(5)
=(q;q)_infty
(6)
=1-q-q^2+q^5+q^7-q^(12)-q^(15)+...
(7)

(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

qf(-q^(24))=q(q^(24))_infty
(8)
=sum_(k=-infty)^(infty)(-1)^kq^((6k+1)^2).
(9)

Ramanujan's phi-function phi(q) is defined by

phi(q)=f(q,q)
(10)
=((-q;q^2)_infty(q^2;q^2)_infty)/((q;q^2)_infty(-q^2;q^2)_infty)
(11)
=((-q,-q)_infty)/((q,-q)_infty)
(12)
=theta_3(0,q)
(13)
=1+2q+2q^4+2q^9+2q^(16)+2q^(25)+...
(14)

(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

phi(e^(-pisqrt(2)))=(Gamma(9/8))/(Gamma(5/4))sqrt((Gamma(1/4))/(2^(1/4)pi))
(16)
phi(e^(-pi))=(pi^(1/4))/(Gamma(3/4)),
(17)

where Gamma(x) is a gamma function.

Ramanujan's psi-function psi(q) is defined by

psi(q)=f(q,q^3)
(18)
=(-q;q)_infty(q^2;q^2)_infty
(19)
=((q^2;q^2)_infty)/((q;q^2)_infty)
(20)
=1/2q^(-1/8)theta_2(0,q^(1/2))
(21)
=sum_(k=0)^(infty)q^(k(k+1)/2)
(22)
=1+q+q^3+q^6+q^(10)+q^(15)+q^(21)+...
(23)

(OEIS A010054; Berndt 1985, p. 37).

Ramanujan's chi-function chi(q) is defined by

chi(q)=(-q;q^2)_infty
(24)
=product_(k=0)^(infty)(1+q^(2k+1))
(25)
=1+q+q^3+q^4+q^5+q^6+q^7+2q^8+...
(26)

(OEIS A000700; Berndt 1985, p. 37).

A different phi function is sometimes defined as

 phi^'(q)=sqrt((theta_2(0,q))/(theta_3(0,q))),
(27)

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 Formula

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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 Functions

Cite this as:

Weisstein, Eric W. "Ramanujan Theta Functions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RamanujanThetaFunctions.html

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