In the last post we saw that using a trigonometric identity Ramanujan was able to express the functions S2n−1(x) or equivalently Φ0,2n−1(x) in terms of simpler functions P(x),Q(x),R(x). Continuing our journey further we start with the equation:
(14cotθ2+xsinθ1−x+x2sin2θ1−x2+x3sin3θ1−x3+⋯)2=(14cotθ2)2+xcosθ(1−x)2+x2cos2θ(1−x2)2+x3cos3θ(1−x3)2+⋯+12{x(1−cosθ)1−x+2x2(1−cos2θ)1−x2+3x3(1−cos3θ)1−x3+⋯}
which was established in the last post.
We show one calculation for Φ2,3(x). Clearly from (9) we have Φ2,3(x)=(xddx)2Φ0,1(x)=xddxΦ1,2(x)=xddx(Q−P2288)=x288(dQdx−2PdPdx)=1288(PQ−R3−P3−PQ6)=3PQ−2R−P31728 Using equations (6),(7),(8) we can see that xddx(Q3−R2)=3Q2xdQdx−2RxdRdx=3Q2PQ−R3−2RPR−Q22=PQ3−RQ2−PR2+RQ2=P(Q3−R2) and hence it follows that xddx{log(Q3−R2)}=P Aagain if we note the definition of eta function η(x) given by η(x)=x1/24(1−x)(1−x2)(1−x3)(1−x4)⋯ it is clear that P=24xddx{log(η(x))} or P=xddx{log(η24(x))} From (10),(11) it follows that we must have Q3−R2=Aη24(x) where A is some constant. Since we have Q=1+240x+⋯,R=1−504x+⋯ it follows that Q3−R2=(3⋅240+2⋅504)x+⋯=1728x+⋯ and therefore the constant A must be 1728.
We finally arrive at the beautiful identity: Q3−R2=1728x{(1−x)(1−x2)(1−x3)⋯}24
Similarly from (26) we get: 11\sigma_{9}(n) = \{21\sigma_{5}(n) - 10\sigma_{3}(n)\} + 5040\sum_{i = 1}^{n - 1}\sigma_{3}(i)\sigma_{5}(n - i) and thus 11\sigma_{9}(n) + 10\sigma_{3}(n) - 21\sigma_{5}(n) is divisible by 5040.
From (25) and (28) we get 1 - 24\sum_{n = 1}^{\infty}\sigma_{13}(n)x^{n} = \left(1 + 480\sum_{n = 1}^{\infty}\sigma_{7}(n)x^{n}\right)\left(1 - 504\sum_{n = 1}^{\infty}\sigma_{5}(n)x^{n}\right) and thus on equating coefficients of x^{n} we get \sigma_{13}(n) = \{21\sigma_{5}(n) - 20\sigma_{7}(n)\} + 10080\sum_{i = 1}^{n - 1}\sigma_{5}(i)\sigma_{7}(n - i) We can rewrite the equation (27) as 691 + 65520\Phi_{0, 11}(x) = 441(Q^{3} - R^{2}) + 691R^{2} so that 691 + 65520\Phi_{0, 11}(x) = 441\cdot 1728\eta^{24}(x) + 691R^{2}\tag{29} Let us define Ramanujan's Tau function \tau(n) by \sum_{n = 1}^{\infty}\tau(n)x^{n} = \eta^{24}(x) = x\prod_{n = 1}^{\infty}(1 - x^{n})^{24} and then equating coefficients of x^{n} in (29) we get \begin{align}65520\sigma_{11}(n) &= 441\cdot 1728\tau(n) - 691\cdot 1008\sigma_{5}(n)\notag\\ &\,\,\,\,+ 691\cdot 504^{2}\sum_{i = 1}^{n - 1}\sigma_{5}(i)\sigma_{5}(n - i)\notag\\ \Rightarrow 65\sigma_{11}(n) &= 756\tau(n) - 691\sigma_{5}(n) + 691\cdot 252\sum_{i = 1}^{n - 1}\sigma_{5}(i)\sigma_{5}(n - i)\notag\\ \Rightarrow 65(\sigma_{11}(n) - \tau(n)) &= 691\tau(n) - 691\sigma_{5}(n) + 691\cdot 252\sum_{i = 1}^{n - 1}\sigma_{5}(i)\sigma_{5}(n - i)\notag\end{align} and thus we arrive at the famous congruence satisfied by \tau(n) \tau(n) \equiv \sigma_{11}(n) \pmod{691}\tag{30}
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Expression of Φ1,2n(x) in terms of P,Q,R
Using the above identity we will now express the functions Φ1,2n(x) in terms of P,Q,R. To do so we need to express both the LHS and RHS of (1) in series of powers of θ. From the last post we can see that 14cotθ2=12θ−B22⋅2⋅θ1!+B42⋅4⋅θ33!−⋯=12θ+E1θ1!−E3θ33!+E5θ55!−⋯xm1−xmsinmθ=xm1−xm(mθ−m3θ33!+m5θ55!−⋯) and therefore the LHS of (1) can be written as: (12θ+θ1!S1(x)−θ33!S3(x)+θ55!S5(x)−⋯)2 For the RHS of (1) we can see that (14cotθ2)2=116cot2θ2=14θ2−124+12(−B42⋅4θ22!+B62⋅6θ44!−B82⋅8θ66!+⋯)=14θ2+E1+12(E3θ22!−E5θ44!+⋯) and mxm(1−cosmθ)1−xm=mxm1−xm(m2θ22!−m4θ44!+⋯)xmcosmθ(1−xm)2=xm(1−xm)2(1−m2θ22!+m4θ44!−⋯) Noting that ∞∑m=1mrxm(1−xm)2=∞∑m=1mrxm∞∑n=1nxmn−m=∞∑m=1∞∑n=1mrnxmn=Φ1,r(x) the RHS of equation (1) can be written as: 14θ2+E1+Φ1,0(x)−θ22!Φ1,2(x)+θ44!Φ1,4(x)−θ66!Φ1,6(x)+⋯+12(θ22!S3(x)−θ44!S5(x)+⋯) Thus the equation (1) is transformed into: (12θ+θ1!S1(x)−θ33!S3(x)+θ55!S5(x)−⋯)2=14θ2+S1(x)−θ22!Φ1,2(x)+θ44!Φ1,4(x)−θ66!Φ1,6(x)+⋯+12(θ22!S3(x)−θ44!S5(x)+⋯) For even positive interger n if we equate the coefficients of θn on both sides of the above equation we get: n+32(n+1)Sn+1(x)−Φ1,n(x)=(n1)S1(x)Sn−1(x)+(n3)S3(x)Sn−3(x)+⋯+(nn−1)Sn−1(x)S1(x) Putting n=2 we get 56S3(x)−Φ1,2(x)=2S21(x)⇒56Q240−Φ1,2(x)=2(−P24)2 or 288Φ1,2(x)=Q−P2 Similarly with n=4,6 we get 720Φ1,4(x)=PQ−R1008Φ1,6(x)=Q2−PR Now that we have expressed Φ1,2n(x) in terms of P,Q,R we proceed to handle the case for general Φr,s(x).Expression of Φr,s(x) in terms of P,Q,R
We know that P=1−24Φ1,0(x)=1−24∞∑n=1σ1(n)xn and therefore xdPdx=−24∞∑n=1nσ1(n)xn=−24Φ1,2(x) and using (3) we get xdPdx=P2−Q12 Similarly we obtain: xdQdx=PQ−R3xdRdx=PR−Q22 Again it is easy to verify (via repeated differentiation) that Φr,s(x)=(xddx)rΦ0,s−r(x) If s−r is odd (i.e. r and s are of different parity) then Φ0,s−r(x) can be expressed as a polynomial in P,Q,R and thereby using (6),(7),(8) we can see from the above equation that Φr,s(x) can be expressed as a polynomial in P,Q,R.We show one calculation for Φ2,3(x). Clearly from (9) we have Φ2,3(x)=(xddx)2Φ0,1(x)=xddxΦ1,2(x)=xddx(Q−P2288)=x288(dQdx−2PdPdx)=1288(PQ−R3−P3−PQ6)=3PQ−2R−P31728 Using equations (6),(7),(8) we can see that xddx(Q3−R2)=3Q2xdQdx−2RxdRdx=3Q2PQ−R3−2RPR−Q22=PQ3−RQ2−PR2+RQ2=P(Q3−R2) and hence it follows that xddx{log(Q3−R2)}=P Aagain if we note the definition of eta function η(x) given by η(x)=x1/24(1−x)(1−x2)(1−x3)(1−x4)⋯ it is clear that P=24xddx{log(η(x))} or P=xddx{log(η24(x))} From (10),(11) it follows that we must have Q3−R2=Aη24(x) where A is some constant. Since we have Q=1+240x+⋯,R=1−504x+⋯ it follows that Q3−R2=(3⋅240+2⋅504)x+⋯=1728x+⋯ and therefore the constant A must be 1728.
We finally arrive at the beautiful identity: Q3−R2=1728x{(1−x)(1−x2)(1−x3)⋯}24
Relation of P,Q,R with Elliptic Integrals K,E
Using the differential equations (6),(7),(8) we can deduce many further properties of P,Q,R. Here we will establish their link with the elliptic integrals K,E and modulus k. We know that dKdk=E−k′2Kkk′2,dEdk=E−Kk Eliminating E from the above relations we get kk′2d2Kdk2+(1−3k2)dKdk−kK=0 Now if we set x=q2 we find that P(q2)=12qddq[log{q1/12(1−q2)(1−q4)(1−q6)⋯}]=12qdkdqddk{log(2−1/3√2Kπ(kk′)1/6)}=24kk′2K2π2{12KdKdk+16k−k6k′2}=(2Kπ)2{3kk′2K⋅E−k′2Kkk′2+k′2−k2}=(2Kπ)2{3EK+k2−2} From (6) and (14)we see that qdP(q2)dq=P2(q2)−Q(q2)6⇒Q(q2)=P2(q2)−6qdP(q2)dq=(2Kπ)4{3EK+k2−2}2−6qdkdqddk{P(q2)}=(2Kπ)4[{3kk′2KdKdk+1−2k2}2−3kk′2K2ddk{3kk′2KdKdk+K2(1−2k2)}]=(2Kπ)4[{3kk′2KdKdk+1−2k2}2−3kk′2K2{3kk′2(Kd2Kdk2+(dKdk)2)+KdKdk(5−13k2)−4kK2}]=(2Kπ)4[{3kk′2KdKdk+1−2k2}2−9k2k′4Kd2Kdk2−9k2k′4K2(dKdk)2−3kk′2K(5−13k2)dKdk+12k2k′2]=(2Kπ)4{−9k2k′4Kd2Kdk2+9kk′2K(3k2−1)dKdk+(1−2k2)2+12k2k′2}=(2Kπ)4{−9kk′2K(kk′2d2Kdk2+(1−3k2)dKdk−kK)+1−4k2+4k4+3k2−3k4}=(2Kπ)4(1−k2+k4) Similarly using(7),(14),(15) we get R(q2)=P(q2)Q(q2)−3q2dQ(q2)dq=(2Kπ)6{3kk′2KdKdk+1−2k2}{1−k2k′2}−3kk′2K2π2ddk{(2Kπ)4{1−k2k′2}}=(2Kπ)6[{3kk′2KdKdk+1−2k2}{1−k2k′2}−3kk′24K4ddk{K4{1−k2k′2}}]=(2Kπ)6[{3kk′2KdKdk+1−2k2}{1−k2k′2}−3kk′24K4{K4(4k3−2k)+4K3dKdk(1−k2k′2)}]=(2Kπ)6[{3kk′2KdKdk+1−2k2}{1−k2k′2}+3k2k′22(1−2k2)−3kk′2KdKdk(1−k2k′2)]=(2Kπ)6(1−2k2)(1+k2k′22)=(2Kπ)6(1+k2)(1−2k2)(1−k22) Going further we can replace q2 by q in (15),(16) and note that this leads to replacing K by (1+k)K and k by 2√k/(1+k) (see Landen's Transformation of second order). When this is done we get: Q(q)=(2(1+k)Kπ)4(1−4k(1+k)2+16k2(1+k)4)=(2Kπ)4{(1+k)4−4k(1+k)2+16k2}=(2Kπ)4{(1+k)2(1−k)2+16k2}=(2Kπ)4(1+14k2+k4) and R(q)=(2(1+k)Kπ)6(1+4k(1+k)2)(1−8k(1+k)2)(1−2k(1+k)2)=(2Kπ)6{(1+k2+6k)(1+k2−6k)(1+k2)}=(2Kπ)6{((1+k2)2−36k2)(1+k2)}=(2Kπ)6(1+k2)(1−34k2+k4) To get P(q) we need to first see how η(q2) transforms when q2 is replaced by q. Clearly we have η(q2)=q1/12(1−q2)(1−q4)(1−q6)⋯=2−1/3√2Kπ(kk′)1/6 so that η(q)=q1/24(1−q)(1−q2)(1−q3)⋯=2−1/3√2(1+k)Kπ(2√k1+k√1−4k(1+k)2)1/6=2−1/6√2Kπk1/12k′1/3 and then we can see that P(q)=24qddq{logη(q)}=48kk′2K2π2ddk{log(2−1/6√2Kπk1/12k′1/3)}=12kk′2(2Kπ)2(12KdKdk+112k−k3k′2)=(2Kπ)2(6kk′2KdKdk+1−5k2)=(2Kπ)2(6EK+k2−5) Another set of formulas come by replacing q with −q in (17),(18),(20). Note that replacing q with −q leads to replacing K with k′K and k2 with (−k2/k′2). With this understanding we have: Q(−q)=(2k′Kπ)4(1−14k2k′2+k4k′4)=(2Kπ)4(k′4−14k2k′2+k4)=(2Kπ)4{(1−k2)2−14k2(1−k2)+k4}=(2Kπ)4(1−16k2+16k4)=(2Kπ)4(1−16k2k′2)=(2Kπ)4(1−4G−24) and R(−q)=(2k′Kπ)6(1−k2k′2)(1+34k2k′2+k4k′4)=(2Kπ)6(1−2k2)(k′4+34k2k′2+k4)=(2Kπ)6(1−2k2)(1+32k2k′2)=(2Kπ)6(1−2k2)(1+8G−24) To get P(−q) requires more work and to that end we first rewrite the relation (20) in another form: P(q)=(2Kπ)2(6kk′2KdKdk+1−5k2)=(2Kπ)2(12k2k′22kKdKdk+1−5k2)=(2Kπ)2(12k2k′2KdKd(k2)+1−5k2) and then we can find P(−q) as follows: P(−q)=(2k′Kπ)2{−12k2k′2(1+k2k′2)1k′Kd(k′K)d(−k2/k′2)+1+5k2k′2}=(2Kπ)2(6kk′Kd(k′K)dk+1+4k2)=(2Kπ)2(6kk′2KdKdk+1−2k2)=(2Kπ)2(6(E−k′2K)K+1−2k2)=(2Kπ)2(6EK+4k2−5) We next define another invariant J(q) by: J(q)=Q3(−q)−R2(−q)Q3(−q)=1728η24(−q)Q3(−q) Now from (19) we see that η24(q)=2−4(2Kπ)12k2(1−k2)4 and hence replacing q by −q we get η24(−q)=2−4(2k′Kπ)12⋅−k2k′2(1+k2k′2)4=2−4(2Kπ)12(−k2k′2)=−2−6(2Kπ)12G−24 Using (21) and definition of J(q) above we arrive at J(q) = \frac{Q^{3}(-q) - R^{2}(-q)}{Q^{3}(-q)} = \frac{-27G^{-24}}{(1 - 4G^{-24})^{3}} = \frac{-27G^{48}}{(G^{24} - 4)^{3}}\tag{24} Also using expressions for \eta^{24}(-q), Q(-q), R(-q) in terms of G we can establish that Q^{3}(-q) - R^{2}(-q) = 1728\eta^{24}(-q) thereby furnishing another proof of (12).Identities Concerning Divisor Functions
From the last post we have the following identities: \begin{align}1 + 480\Phi_{0, 7}(x) &= Q^{2}\tag{25}\\ 1 - 264\Phi_{0, 9}(x) &= QR\tag{26}\\ 691 + 65520\Phi_{0, 11}(x) &= 441Q^{3} + 250R^{2}\tag{27}\\ 1 - 24\Phi_{0, 13}(x) &= Q^{2}R\tag{28}\end{align} First of these identities can be written as: 1 + 480\sum_{n = 1}^{\infty}\sigma_{7}(n)x^{n} = \left(1 + 240\sum_{n = 1}^{\infty}\sigma_{3}(n)x^{n}\right)^{2} Equating coefficients of x^{n} on both sides we get: \sigma_{7}(n) = \sigma_{3}(n) + 120\sum_{i = 1}^{n - 1}\sigma_{3}(i)\sigma_{3}(n - i) so that \sigma_{7}(n) - \sigma_{3}(n) is divisible by 120 for all positive integers n.Similarly from (26) we get: 11\sigma_{9}(n) = \{21\sigma_{5}(n) - 10\sigma_{3}(n)\} + 5040\sum_{i = 1}^{n - 1}\sigma_{3}(i)\sigma_{5}(n - i) and thus 11\sigma_{9}(n) + 10\sigma_{3}(n) - 21\sigma_{5}(n) is divisible by 5040.
From (25) and (28) we get 1 - 24\sum_{n = 1}^{\infty}\sigma_{13}(n)x^{n} = \left(1 + 480\sum_{n = 1}^{\infty}\sigma_{7}(n)x^{n}\right)\left(1 - 504\sum_{n = 1}^{\infty}\sigma_{5}(n)x^{n}\right) and thus on equating coefficients of x^{n} we get \sigma_{13}(n) = \{21\sigma_{5}(n) - 20\sigma_{7}(n)\} + 10080\sum_{i = 1}^{n - 1}\sigma_{5}(i)\sigma_{7}(n - i) We can rewrite the equation (27) as 691 + 65520\Phi_{0, 11}(x) = 441(Q^{3} - R^{2}) + 691R^{2} so that 691 + 65520\Phi_{0, 11}(x) = 441\cdot 1728\eta^{24}(x) + 691R^{2}\tag{29} Let us define Ramanujan's Tau function \tau(n) by \sum_{n = 1}^{\infty}\tau(n)x^{n} = \eta^{24}(x) = x\prod_{n = 1}^{\infty}(1 - x^{n})^{24} and then equating coefficients of x^{n} in (29) we get \begin{align}65520\sigma_{11}(n) &= 441\cdot 1728\tau(n) - 691\cdot 1008\sigma_{5}(n)\notag\\ &\,\,\,\,+ 691\cdot 504^{2}\sum_{i = 1}^{n - 1}\sigma_{5}(i)\sigma_{5}(n - i)\notag\\ \Rightarrow 65\sigma_{11}(n) &= 756\tau(n) - 691\sigma_{5}(n) + 691\cdot 252\sum_{i = 1}^{n - 1}\sigma_{5}(i)\sigma_{5}(n - i)\notag\\ \Rightarrow 65(\sigma_{11}(n) - \tau(n)) &= 691\tau(n) - 691\sigma_{5}(n) + 691\cdot 252\sum_{i = 1}^{n - 1}\sigma_{5}(i)\sigma_{5}(n - i)\notag\end{align} and thus we arrive at the famous congruence satisfied by \tau(n) \tau(n) \equiv \sigma_{11}(n) \pmod{691}\tag{30}
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