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Ramanujan Theta Functions

المؤلف:  Berndt, B. C.; Huang, S.-S.; Sohn, J.; and Son, S. H.

المصدر:  "Some Theorems on the Rogers-Ramanujan Continued Fraction in Ramanujan,s Lost Notebook." Trans. Amer. Math. Soc. 352

الجزء والصفحة:  ...

31-8-2019

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Ramanujan Theta Functions

 

Ramanujan's two-variable theta function  is defined by

 

(1)

for  (Berndt 1985, p. 34; Berndt et al. 2000). It satisfies

(2)

and

			(3)
			(4)

(Berndt 1985, pp. 34-35; Berndt et al. 2000), where  is a q-Pochhammer symbol, i.e., a q-series.

A one-argument form of  is also defined by

			(5)
			(6)
			(7)

(OEIS A010815; Berndt 1985, pp. 36-37; Berndt et al. 2000), where  is a q-Pochhammer symbol. The identities above are equivalent to the pentagonal number theorem.

The function also satisfies

			(8)
			(9)

Ramanujan's -function  is defined by

			(10)
			(11)
			(12)
			(13)
			(14)

(OEIS A000122), where  is a Jacobi theta function (Berndt 1985, pp. 36-37).  is a generalization of , with the two being connected by

(15)

Special values of  include

			(16)
			(17)

where  is a gamma function.

Ramanujan's -function  is defined by

			(18)
			(19)
			(20)
			(21)
			(22)
			(23)

(OEIS A010054; Berndt 1985, p. 37).

Ramanujan's -function  is defined by

			(24)
			(25)
			(26)

(OEIS A000700; Berndt 1985, p. 37).

A different  function is sometimes defined as

(27)

where  is again a Jacobi theta function, which has special value

(28)

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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."