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Series solutions for a large family of Dyson–Schwinger equations are given as expansions over decorated rooted connected chord diagrams. The analytic input to the new expansions are the expansions of the regularized integrals for the primitive graphs building the Dyson–Schwinger equation. Each decorated chord diagram contributes a weighted monomial in the coefficients of the expansions of the primitives and so indexes the analytic solution in a tightly controlled way.
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Markus Hihn, Karen Yeats, Generalized chord diagram expansions of Dyson–Schwinger equations. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 6 (2019), no. 4, pp. 573–605DOI 10.4171/AIHPD/79