In this paper the local limit theorem for lattice distributions has been applied to deduce the asymptotic behavior (as n→∞\documentclass[12pt]{minimal}
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\begin{document}$$n\rightarrow \infty $$\end{document}) for the modified Bessel function Iν-βn(2xλαn)\documentclass[12pt]{minimal}
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\begin{document}$$I_{\nu -\beta _n}( 2x\sqrt{\lambda \alpha _n})$$\end{document}, where (αn)n∈N\documentclass[12pt]{minimal}
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\begin{document}$$(\alpha _n)_{n\in {\mathbb {N}}}$$\end{document} and (βn)n∈N\documentclass[12pt]{minimal}
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\begin{document}$$(\beta _n)_{n\in {\mathbb {N}}}$$\end{document} are any two increasing sequences of natural numbers such that (λαn-βn)22λαn→μλ≥0\documentclass[12pt]{minimal}
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\begin{document}$$\frac{(\lambda \alpha _n-\beta _n)^2}{2\lambda \alpha _n}\rightarrow \mu _\lambda \ge 0$$\end{document} as n→∞\documentclass[12pt]{minimal}
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\begin{document}$$n\rightarrow \infty $$\end{document}, λ>0,\documentclass[12pt]{minimal}
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\begin{document}$$\lambda >0,$$\end{document} and ν≥0.\documentclass[12pt]{minimal}
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\begin{document}$$\nu \ge 0.$$\end{document} Our asymptotics is uniformly valid in the compact subsets of (0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$(0,\infty )$$\end{document}.
