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**A generalization of Kummer's identity**

**Abstract**

The well-known Kummer's formula evaluates the hypergeometric series ${}_2F_1(a,b;c;-1)$ when the relation $b-a+c=1$ holds. In the talk I would present a formula which evaluates this series in case when $b-a+c$ is an integer. The formula expresses the series as a linear combination of two $\Gamma$ terms, with coefficients being hypergeometric ${}_3F_2$ polynomials. Using a standard transformation like 2.9(3) from Bateman's "Higher transcendental functions" one obtains a similar generalization of the identities of Bailey and Gauss which evaluate some ${}_2F_1$ series at $1/2$.