Characterization of polyanalytic functions by meromorphic extensions from chains of circles

Let C_t = {z ∈ ℂ: {pipe}z - c(t){pipe} = r(t), t ∈ (0, 1)} be a C¹-family of circles in the plane such that lim_t→0+C_t = {a}, lim_t→1-C_t = {b}, a ≠ b, and {pipe}c′(t){pipe}² + {pipe}r′(t){pipe}² ≠ 0. The discriminant set S of the family is defined as the closure of the set {c(t) + r(t)w(t), t ∈ [0, 1]}, where w = w(t) is the root of the quadratic equation c̄′(t)w² + 2r′(t)w + c′(t) = 0 with {pipe}w{pipe} < 1, if such a root exists. Suppose that S does not contain a continuous curve joining a and b. We prove that if a smooth and regular (in a certain sense) function f in Ω = ∪C_t, possesses, for each t ∈ (0, 1), a meromorphic extension inside C_t whose only singular point is a pole at c(t) of order at most ν ∈ {0} ∪ ℕ, then f is polyanalytic of order at most ν, i. e., f(z) = h₀(z) + h₁(z)z̄+̇̇̇z̄^νh_ν(z), where the h′_js are analytic functions in Ω. For ν = 0, the condition on S can be omitted. If {pipe}c′(t){pipe} > {pipe}r′(t){pipe}, t ∈ (0, 1), then S = ∅ and the condition of regularity can be dropped. A hyperbolic version of the result is also given.