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The objective of this paper is to establish a new relationship between theVeronese subsequent analytic solutions of the Euclidean $\mathbb{C}P^{2s}$sigma model in two dimensions and the orthogonal Krawtchouk polynomials. Weshow that such solutions of the $\mathbb{C}P^{2s}$ model, defined on theRiemann sphere and having a finite action, can be explicitly parametrised interms of these polynomials. We apply the obtained results to the analysis ofsurfaces associated with $\mathbb{C}P^{2s}$ sigma models, defined using thegeneralized Weierstrass formula for immersion. We show that these surfaces arespheres immersed in the $\mathfrak{su}(2s+1)$ Lie algebra, and express severalother geometrical characteristics in terms of the Krawtchouk polynomials.Finally, a new connection between the $\mathfrak{su}(2)$ spin-s representationand the $\mathbb{C}P^{2s}$ model is explored in detail. It is shown that forany given holomorphic vector function in $\mathbb{C}^{2s+1}$ written as aVeronese sequence, it is possible to derive subsequent solutions of the$\mathbb{C}P^{2s}$ model through algebraic recurrence relations which turn outto be simpler than the analytic relations known in the literature.

Veronese subsequent analytic solutions of the $\mathbb{C}P^{2s}$ sigma model equations described via Krawtchouk polynomials