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Ramanujan change of base

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1: 20.11 Generalizations and Analogs
§20.11(iii) Ramanujan’s Change of Base
These results are called Ramanujan’s changes of base. …
2: 8.11 Asymptotic Approximations and Expansions
If x = a + ( 2 a ) 1 2 y and a + , then
8.11.11 γ * ( 1 - a , - x ) = x a - 1 ( - cos ( π a ) + sin ( π a ) π ( 2 π F ( y ) + 2 3 2 π a ( 1 - y 2 ) ) e y 2 + O ( a - 1 ) ) ,
8.11.14 e n x = e n ( n x ) + ( n x ) n n ! S n ( x ) ,
8.11.15 S n ( x ) = γ ( n + 1 , n x ) ( n x ) n e - n x .
3: Bibliography B
  • R. J. Baxter (1981) Rogers-Ramanujan identities in the hard hexagon model. J. Statist. Phys. 26 (3), pp. 427–452.
  • A. Berkovich and B. M. McCoy (1998) Rogers-Ramanujan Identities: A Century of Progress from Mathematics to Physics. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 163–172.
  • B. C. Berndt, S. Bhargava, and F. G. Garvan (1995) Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (11), pp. 4163–4244.
  • B. C. Berndt (1989) Ramanujan’s Notebooks. Part II. Springer-Verlag, New York.
  • B. C. Berndt (1991) Ramanujan’s Notebooks. Part III. Springer-Verlag, Berlin-New York.
  • 4: 15.8 Transformations of Variable
    With ζ = e 2 π i / 3 ( 1 - z ) / ( z - e 4 π i / 3 )
    15.8.32 ( 1 - z 3 ) a ( - z ) 3 a ( 1 Γ ( a + 2 3 ) Γ ( 2 3 ) F ( a , a + 1 3 2 3 ; z - 3 ) + e 1 3 π i z Γ ( a ) Γ ( 4 3 ) F ( a + 1 3 , a + 2 3 4 3 ; z - 3 ) ) = 3 3 2 a + 1 2 e 1 2 a π i Γ ( a + 1 3 ) ( 1 - ζ ) a 2 π Γ ( 2 a + 2 3 ) ( - ζ ) 2 a F ( a + 1 3 , 3 a 2 a + 2 3 ; ζ - 1 ) , | z | > 1 , | ph ( - z ) | < 1 3 π .
    Ramanujan’s Cubic Transformation