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On certain multi-variable rational identities derived from the rigidity of signature of manifolds. (English) Zbl 1379.57041
Summary: R. Song [Sci. China, Ser. A 48, No. 12, 1637–1645 (2005; Zbl 1217.58015)] derives certain multi-variable rational identities by studying torus actions on some homogeneous manifolds and applying the Atiyah-Bott-Segal-Singer Lefschetz fixed point theorem. In this paper, we give a direct proof of these rational identities by using the \(q\)-Lucas theorem. Moreover, we also give a similar new rational identity.
MSC:
57S15 Compact Lie groups of differentiable transformations
58J26 Elliptic genera
58J20 Index theory and related fixed-point theorems on manifolds
11F27 Theta series; Weil representation; theta correspondences
05A19 Combinatorial identities, bijective combinatorics
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