The Equation y′ = wy and the Meromorphic Products

Let K be a complete ultrametric algebraically closed field of characteristic zero and let D be a clopen bounded infraconnected set whose holes have diameter greater than a positive constant p.

Let H(D) be the Banach algebra of the Analytic Elements in D.

The Differential Equation ℓ ( w ) y 1 = w y ( w ∈ H ( D ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/3ce7aa8c-759e-436c-a799-e22f41c5a790/content/eq1937.tif"/> has non zero solutions in H(D) if and only if w is of the form(1) ∑ i = 1 q u i x − c i + ∑ n = 1 ∞ ( 1 x − a n − 1 x − b n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/3ce7aa8c-759e-436c-a799-e22f41c5a790/content/eq1938.tif"/> where the c_i belong to different holes T_i , of D, each u_i , is the residue of w in T_i , and the sequences (a_n),(b_n) satisfy conditions like lim n → ∞ ( a n − b n ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/3ce7aa8c-759e-436c-a799-e22f41c5a790/content/eq1939.tif"/> . The solutions then are in the form λ ∏ i = 1 q ( x − c i ) u i ∏ n = 1 ∞ ( x − a n x − b n ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/3ce7aa8c-759e-436c-a799-e22f41c5a790/content/eq1940.tif"/> .

Then we characterize the w which have integrals in H(D) as the elements of the form 1 p h ∑ n = 1 ∞ 1 x − a n − 1 x − b n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/3ce7aa8c-759e-436c-a799-e22f41c5a790/content/eq1941.tif"/> and we define and study the quasi-integrable elements.

Let B be the open unit disk, and let θ ∈ H(B). We assume ℓ(θ) to have a non zero solution f analytic function in B (not supposed to belong to H(B)). Then f N ∈ H ( B ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/3ce7aa8c-759e-436c-a799-e22f41c5a790/content/eq1942.tif"/> for some N ∈ ℕ if and only if θ is of the form jjU) where uo is of the form 1 N w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/3ce7aa8c-759e-436c-a799-e22f41c5a790/content/eq1943.tif"/> where w (1) above. As a corollary we have an immediate demonstation to show that a function 1 + x N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/3ce7aa8c-759e-436c-a799-e22f41c5a790/content/eq1944.tif"/> does not belong to H(B). When K = ℂ_p, we apply these results to the search of a Frobenius Structure on ℓ(w).