# # Multiplicative identities for the function # eta(x) = prod( n=1, infinity, 1-x^n ) # # A line # r: EXPR # says that # prod(j=1,r-1,if ( gcd(j,r)==1, eta(w^j*x), 1)) = EXPR # where in EXPR eta(x^k) is abbreviated as E(k). #. # Generated by Joerg Arndt, 2009-August-21 # ## eta(-x) == eta(x^2)^3 / ( eta(x) * eta(x^4) ): 2: ( E(2)^3 ) / ( E(1)^1 E(4)^1 ) ## eta(w*x) * eta(w^2*x) (where w=exp(2*Pi*I/3)) == eta(x^3) / ( eta(x) * eta(x^9) ): 3: ( E(3)^4 ) / ( E(1)^1 E(9)^1 ) ## eta(+I*x) * eta(-I*x) == eta(x^4)^8 / ( eta(x^2)^3 * eta(x^8)^3 ): 4: ( E(4)^8 ) / ( E(2)^3 E(8)^3 ) 5: ( E(5)^6 ) / ( E(1)^1 E(25)^1 ) 6: ( E(1)^1 E(4)^1 E(6)^12 E(9)^1 E(36)^1 ) / ( E(2)^3 E(3)^4 E(12)^4 E(18)^3 ) 7: ( E(7)^8 ) / ( E(1)^1 E(49)^1 ) 8: ( E(8)^18 ) / ( E(4)^7 E(16)^7 ) 9: ( E(9)^14 ) / ( E(3)^4 E(27)^4 ) 10: ( E(1)^1 E(4)^1 E(10)^18 E(25)^1 E(100)^1 ) / ( E(2)^3 E(5)^6 E(20)^6 E(50)^3 ) 11: ( E(11)^12 ) / ( E(1)^1 E(121)^1 ) 12: ( E(2)^3 E(8)^3 E(12)^32 E(18)^3 E(72)^3 ) / ( E(4)^8 E(6)^12 E(24)^12 E(36)^8 ) 13: ( E(13)^14 ) / ( E(1)^1 E(169)^1 ) 14: ( E(1)^1 E(4)^1 E(14)^24 E(49)^1 E(196)^1 ) / ( E(2)^3 E(7)^8 E(28)^8 E(98)^3 ) 15: ( E(1)^1 E(9)^1 E(15)^24 E(25)^1 E(225)^1 ) / ( E(3)^4 E(5)^6 E(45)^6 E(75)^4 ) 16: ( E(16)^38 ) / ( E(8)^15 E(32)^15 ) 17: ( E(17)^18 ) / ( E(1)^1 E(289)^1 ) 18: ( E(3)^4 E(12)^4 E(18)^42 E(27)^4 E(108)^4 ) / ( E(6)^12 E(9)^14 E(36)^14 E(54)^12 ) 19: ( E(19)^20 ) / ( E(1)^1 E(361)^1 ) 20: ( E(2)^3 E(8)^3 E(20)^48 E(50)^3 E(200)^3 ) / ( E(4)^8 E(10)^18 E(40)^18 E(100)^8 ) 21: ( E(1)^1 E(9)^1 E(21)^32 E(49)^1 E(441)^1 ) / ( E(3)^4 E(7)^8 E(63)^8 E(147)^4 ) 22: ( E(1)^1 E(4)^1 E(22)^36 E(121)^1 E(484)^1 ) / ( E(2)^3 E(11)^12 E(44)^12 E(242)^3 ) 23: ( E(23)^24 ) / ( E(1)^1 E(529)^1 ) 24: ( E(4)^7 E(16)^7 E(24)^72 E(36)^7 E(144)^7 ) / ( E(8)^18 E(12)^28 E(48)^28 E(72)^18 ) 25: ( E(25)^32 ) / ( E(5)^6 E(125)^6 ) 26: ( E(1)^1 E(4)^1 E(26)^42 E(169)^1 E(676)^1 ) / ( E(2)^3 E(13)^14 E(52)^14 E(338)^3 ) 27: ( E(27)^44 ) / ( E(9)^13 E(81)^13 ) 28: ( E(2)^3 E(8)^3 E(28)^64 E(98)^3 E(392)^3 ) / ( E(4)^8 E(14)^24 E(56)^24 E(196)^8 ) 29: ( E(29)^30 ) / ( E(1)^1 E(841)^1 ) 30: ( E(2)^3 E(3)^4 E(5)^6 E(12)^4 E(18)^3 E(20)^6 E(30)^72 E(45)^6 E(50)^3 E(75)^4 E(180)^6 E(300)^4 E(450)^3 ) / ( E(1)^1 E(4)^1 E(6)^12 E(9)^1 E(10)^18 E(15)^24 E(25)^1 E(36)^1 E(60)^24 E(90)^18 E(100)^1 E(150)^12 E(225)^1 E(900)^1 ) 31: ( E(31)^32 ) / ( E(1)^1 E(961)^1 ) 32: ( E(32)^78 ) / ( E(16)^31 E(64)^31 ) 33: ( E(1)^1 E(9)^1 E(33)^48 E(121)^1 E(1089)^1 ) / ( E(3)^4 E(11)^12 E(99)^12 E(363)^4 )