[MUSIC] Certainly we cannot define exponential function on D in a modular way because D to the power k is not modular operator. I propose the following construction. We consider several modular differential operator. Remind you then Dk. By definition is D + 2kG2 and the operator maps the space of modular form of weight k into the space of modular form of weight k + 2. Now we start with Mk, then we can use Dk against the space Mk + 2. Then we'll use modular differentiator Dk+ 2 and so on. See, the space Mk + 2n is the last operator is Dk + 2n- 2. The result. The composition of M modular differential operator, we denote by Dk, n. Do we have a formula for this operator? This formula certainly is rather complicated and we don't need it. I would like to calculate this operator, a modular modular form. Or it's better to say I would like to calculate. It's major term of order M. Let me explain this idea. When we analyze the action of the differential operator on the quasimodular Eisenstein series G2, we found then a reminder of the definition of this operator for D2. D2, this is the modular operator, which transforms this space of quasimodular form generated by G2 in the space of modular form of weight. For more exactly D, D2 (G2) = D(G2) + 2 times G2 times G2. Because in the modular differential operator, we have the separator of multiplication by G2. The result is up to the constant, the Eisenstein series of weight 4. But I don't care on this constant. I would like to rewrite this identity as follows, D(G2) is congruent to -2G2 to the square modular M4. And now I would like to make the calculation. So we calculate. DK M modular form, modular forms. And I would like to control only the measure term of order M. Now I can formulate the result, the theorem. The major term of this differential operator DKM on the N ends iteration on the modular differential operator is EKN equal to the sum new from 0 til N, N factorial (new factorial N- mu) factorial as a denominal coefficient. Then we'll have the quotient of two gamma function. Gamma in (k+n) / gamma of (mu+n) as in the product of the quasimodular Eisenstein series to the power n minus mu. And then the differential operator to the power mu. So this operator transforms the space of modular form of weight k into the space of modular form of weight K + 2m. Please analyze the formula. [SOUND] In this iteration, we'll have to calculate. A lot of images of this type, and we calculate them using only this congruence. In principle, you can forget about this discussion that this function E is a major term of the differential operator DKM because now we have to prove that this is a modular differential operator. And we can prove it using the induction proof. Induction by the index n. And in this induction, I would like to repeat it. I'm going to use another color. We use the only relation in d G2 is congruent to -2 G2 -- 2C to the square plus G2 times D modular M star. This is the only relation which we use. And using this relation you can prove this is elementary calculation, E k n + 1 = d the modular operator of weight k + 2 n the action of this modular differential at the rate on this pace, M K + 2n. That means on Ekn. And then we have to correct this term but minus 10 / 6 the Eisenstein series of weight 4 internal and then e k n- 1. So you can check this identity by this elementary calculation. Because maybe D k plus 2 n = D + 2 k + 2 n multiplication by G2, so you have to act. By D on all Eisenstein series in our function. So the only identity, I would like to repeat this, only one congruence gives us this identity. And this gives us a theorem. And now. How to simplify this operator? Certainly in our operator, you see the convolution of the differential operator. D ends the powers of the quasimodular Eisenstein series. So now I would like to define the following formal power series. E, this is analog of the exponent of the differential operator, in x is equal to the sum n greater to 0. I would like, first of all, to divide. The operator e k n by the numerator. Then, I have e K, n x to the pow of n. And certainly, if we. Divide by the numerator. The numerator. We can write this formal power series is a product of E to the power 2 G two times x. And then, the next operator, the sum. The rest will be. 1 over new factorial gamma Nu + n. In the numerator will have differential operator D nu, the sum for all nu x to the power nu. So, we've proved this identity which I would like to analyze now very carefully. Let me repeat the fact we proved. First of all, we proved the theorem then the correspondent. The major part of the iteration is a modular operator. Then we write down the formal, the composition of the formal power series of our quasi exponential function, or it's better to say function, of their differential operator. [MUSIC]