Notes on Entry 27 of Chapter 25 of Ramanujan's Notebooks, Part IV From Michael Somos, Dec 08 2009 This refers to Bruce C. Berndt, Ramanujan's Notebooks, Part IV, Springer-Verlag, 1984 In this case, Ramanujan made a rare innocent copying error. He knew exactly what was the correct identity but forgot to copy one term on the left side. Sequences 1, 2, 3, 4 mentioned below are entries A170773, A170770, A170771 and A170772 respectively in the OEIS. /* entry27.gp -- Ramanujan Entry 27 */ \\ 08 Dec 2009 Michael Somos \\ check an identity chk(ex)=if(eval(ex)!=1,print(ex" false")) \\ Ramanujan's f(-q^n) et(n,oo=oo)=eta(q^n+q*O(q^oo)); \\ Ramanujan's phi(q^n) rphi(n,oo=oo)=et(2*n,oo)^5/(et(n,oo)*et(4*n,oo))^2; \\ Ramanujan's psi(q^n) rpsi(n,oo=oo)=et(2*n,oo)^2/et(n,oo); \\ product of several phi(q^n) factors fv(vn,oo=oo)=prod(k=1,#vn,rphi(vn[k],oo)); \\ product of several psi(q^n) factors gv(vn,oo=oo)=prod(k=1,#vn,rpsi(vn[k],oo)); oo=600; \\ our infinity \\ Part IV page 170 Entry 27 Equation (27.1) left side sum L1=fv([1,7,9,63]); L2=subst(L1,q,-q); L3=4*q^4*et(6)^2*et(42)^2; \\ Ramanujan missed this term accidently (not in Berndt!!) L4=16*q^20*gv([2,14,18,126]); L=(L1+L2+L3+L4)/2; \\ Part IV page 170 Entry 27 Equation (27.1) right side sum R1=fv([1,63]); R2=subst(R1,q,-q); R3=4*q^16*gv([2,126]); R=(R1+R2+R3)/2; \\ This validates the corrected identity chk("L==R^2"); print();print("sequence 1");print(); print("Expansion of ( phi(q) * phi(q^7) * phi(q^9) * phi(q^63) + phi(-q) * phi(-q^7) * phi(-q^9) * phi(-q^63) + 4 * q^4 * f(-q^6)^2 * f(-q^42)^2 + 16 * q^20 * psi(q^2) * psi(q^14) * psi(q^18) * psi(q^126) ) / 2 in powers of q^2 where phi(), psi(), and f() are Ramanujan theta functions."); print();print(L+O(q^32)); print();for(n=0,100,print(n" "polcoeff(L,2*n))); print();print("sequence 2");print(); print("Expansion of ( phi(q) * phi(q^63) + phi(-q) * phi(-q^63) + 4 * q^16 * psi(q^2) * psi(q^126) ) / 2 in powers of q^2 where phi(), psi() are Ramanujan theta functions."); print();print(R+O(q^64)); print();for(n=0,100,print(n" "polcoeff(R,2*n))); \\ Note that sequence 1 is convolution square of sequence 2 \\ Part IV page 171 incorrect left side q-series Lside= 2*(L1+L2+L3); \\ incorrect left side print();print("sequence 3");print(); print("Expansion of 2 * ( phi(q) * phi(q^7) * phi(q^9) * phi(q^63) + phi(-q) * phi(-q^7) * phi(-q^9) * phi(-q^63) + 4 * q^4 * f(-q^6)^2 * f(-q^42)^2 ) in powers of q^2 where phi(), psi(), and f() are Ramanujan theta functions."); print();print(Lside+O(q^34)); \\ compare with text \\ q-series expansion in powers of q^2 print();for(n=0,100,print(n" "polcoeff(Lside,2*n))); \\ Part IV page 171 correct right side q-series Rside= (R1+R2+R3)^2; \\ correct right side print();print("sequence 4");print(); print("Expansion of ( phi(q) * phi(q^63) + phi(-q) * phi(-q^63) + 4 * q^16 * psi(q^2) * psi(q^126) ) ^ 2 in powers of q^2 where phi(), psi() are Ramanujan theta functions."); print();print(Rside+O(q^32)); \\ compare with text \\ q-series expansion in powers of q^2 print();for(n=0,100,print(n" "polcoeff(Rside,2*n))); GP/PARI CALCULATOR Version 2.4.3 (development svn-11973M) i686 running linux (ix86 kernel) 32-bit version -- debugging compiled: Sep 30 2009, gcc-3.0.4 (readline v5.0 enabled, extended help enabled) Copyright (C) 2000-2008 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?12 for how to get moral (and possibly technical) support. parisize = 4000000, primelimit = 500509 sequence 1 Expansion of ( phi(q) * phi(q^7) * phi(q^9) * phi(q^63) + phi(-q) * phi(-q^7) * phi(-q^9) * phi(-q^63) + 4 * q^4 * f(-q^6)^2 * f(-q^42)^2 + 16 * q^20 * psi(q^2) * psi(q^14) * psi(q^18) * psi(q^126) ) / 2 in powers of q^2 where phi(), psi(), and f() are Ramanujan theta functions. 1 + 4*q^4 + 4*q^8 + 8*q^16 + 4*q^18 + 16*q^20 + 12*q^22 + 8*q^26 + 4*q^28 + O(q^32) 0 1 1 0 2 4 3 0 4 4 5 0 6 0 7 0 8 8 9 4 10 16 11 12 12 0 13 8 14 4 15 0 16 24 17 16 18 12 19 16 20 24 21 0 22 20 23 12 24 0 25 16 26 32 27 16 28 4 29 20 30 0 31 24 32 36 33 0 34 32 35 0 36 28 37 24 38 32 39 0 40 56 41 40 42 0 43 32 44 60 45 24 46 52 47 40 48 0 49 0 50 76 51 0 52 64 53 28 54 48 55 56 56 8 57 0 58 60 59 32 60 0 61 48 62 56 63 4 64 84 65 48 66 0 67 48 68 88 69 0 70 16 71 36 72 60 73 40 74 80 75 0 76 96 77 12 78 0 79 48 80 144 81 52 82 72 83 56 84 0 85 80 86 80 87 0 88 124 89 56 90 72 91 8 92 108 93 0 94 88 95 96 96 0 97 56 98 4 99 48 100 140 sequence 2 Expansion of ( phi(q) * phi(q^63) + phi(-q) * phi(-q^63) + 4 * q^16 * psi(q^2) * psi(q^126) ) / 2 in powers of q^2 where phi(), psi() are Ramanujan theta functions. 1 + 2*q^4 + 4*q^16 + 2*q^18 + 2*q^22 + 2*q^28 + 4*q^36 + 2*q^46 + 2*q^58 + O(q^64) 0 1 1 0 2 2 3 0 4 0 5 0 6 0 7 0 8 4 9 2 10 0 11 2 12 0 13 0 14 2 15 0 16 0 17 0 18 4 19 0 20 0 21 0 22 0 23 2 24 0 25 0 26 0 27 0 28 0 29 2 30 0 31 0 32 6 33 0 34 0 35 0 36 6 37 0 38 0 39 0 40 0 41 0 42 0 43 0 44 6 45 0 46 0 47 0 48 0 49 0 50 2 51 0 52 0 53 2 54 0 55 0 56 4 57 0 58 0 59 0 60 0 61 0 62 0 63 2 64 0 65 0 66 0 67 0 68 0 69 0 70 0 71 2 72 8 73 0 74 4 75 0 76 0 77 2 78 0 79 0 80 0 81 2 82 0 83 0 84 0 85 0 86 4 87 0 88 0 89 0 90 0 91 0 92 6 93 0 94 0 95 0 96 0 97 0 98 2 99 4 100 0 sequence 3 Expansion of 2 * ( phi(q) * phi(q^7) * phi(q^9) * phi(q^63) + phi(-q) * phi(-q^7) * phi(-q^9) * phi(-q^63) + 4 * q^4 * f(-q^6)^2 * f(-q^42)^2 ) in powers of q^2 where phi(), psi(), and f() are Ramanujan theta functions. 4 + 16*q^4 + 16*q^8 + 32*q^16 + 16*q^18 + 32*q^20 + 16*q^22 + 16*q^28 + 64*q^32 + O(q^34) 0 4 1 0 2 16 3 0 4 16 5 0 6 0 7 0 8 32 9 16 10 32 11 16 12 0 13 0 14 16 15 0 16 64 17 32 18 16 19 32 20 0 21 0 22 48 23 16 24 0 25 0 26 96 27 0 28 16 29 16 30 0 31 32 32 48 33 0 34 64 35 0 36 80 37 32 38 32 39 0 40 128 41 64 42 0 43 64 44 208 45 32 46 112 47 32 48 0 49 0 50 240 51 0 52 160 53 48 54 128 55 96 56 32 57 0 58 144 59 32 60 0 61 96 62 96 63 16 64 208 65 64 66 0 67 64 68 288 69 0 70 32 71 48 72 144 73 32 74 192 75 0 76 224 77 16 78 0 79 64 80 416 81 80 82 160 83 64 84 0 85 128 86 128 87 0 88 336 89 64 90 160 91 0 92 208 93 0 94 224 95 128 96 0 97 32 98 16 99 64 100 368 sequence 4 Expansion of ( phi(q) * phi(q^63) + phi(-q) * phi(-q^63) + 4 * q^16 * psi(q^2) * psi(q^126) ) ^ 2 in powers of q^2 where phi(), psi() are Ramanujan theta functions. 4 + 16*q^4 + 16*q^8 + 32*q^16 + 16*q^18 + 64*q^20 + 48*q^22 + 32*q^26 + 16*q^28 + O(q^32) 0 4 1 0 2 16 3 0 4 16 5 0 6 0 7 0 8 32 9 16 10 64 11 48 12 0 13 32 14 16 15 0 16 96 17 64 18 48 19 64 20 96 21 0 22 80 23 48 24 0 25 64 26 128 27 64 28 16 29 80 30 0 31 96 32 144 33 0 34 128 35 0 36 112 37 96 38 128 39 0 40 224 41 160 42 0 43 128 44 240 45 96 46 208 47 160 48 0 49 0 50 304 51 0 52 256 53 112 54 192 55 224 56 32 57 0 58 240 59 128 60 0 61 192 62 224 63 16 64 336 65 192 66 0 67 192 68 352 69 0 70 64 71 144 72 240 73 160 74 320 75 0 76 384 77 48 78 0 79 192 80 576 81 208 82 288 83 224 84 0 85 320 86 320 87 0 88 496 89 224 90 288 91 32 92 432 93 0 94 352 95 384 96 0 97 224 98 16 99 192 100 560