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In this paper, we derive a local expression of the standard $L$-function attached to a Jacobi form of higher degree in terms of a certain power series related to its Fourier coefficients. This can be regarded as an analogue of Andrianov's identity for Siegel modular forms. As an application, we also show the rationality theorem for a formal power series related to a polynomial appearing in the theory of local densities of quadratic forms, which is very similar to the result obtained by Boecherer and Sato. This would play an important role in proving a conjecture on the period of the Duke-Imamoglu-Ikeda lift proposed by Ikeda.

On the Andrianov-type identity for power series attached to Jacobi forms and its application