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In quantum mechanics, bosonic operators are mathematical objects that areused to represent the creation ($a^\dagger$) and annihilation ($a$) of bosonicparticles. The natural power of a linear combination of bosonic operatorsrepresents an operator $(a^\dagger x+ay)^n$ with $n$ as the exponent and$x,\,y$ are the variables free from bosonic operators. The normal ordering ofthese operators is a mathematical technique that arranges the operators so thatall the creation operators are to the left of the annihilation operators,reducing the number of terms in the expression. In this paper, we present ageneral expansion of the natural power of a linear combination of bosonicoperators in normal order. We show that the expansion can be expressed in termsof binomial coefficients and the product of the normal-ordered operators usingthe direct method and than prove it using the fundamental principle ofmathematical induction. We also derive a formula for the coefficients of theexpansion in terms of the number of bosons and the commutation relation betweenthe creation and annihilation operators. Our results have importantapplications in the study of many-body systems in quantum mechanics, such as inthe calculation of correlation functions and the evaluation of the partitionfunction. The general expansion presented in this paper provides a powerfultool for analyzing and understanding the behavior of bosonic systems, and canbe applied to a wide range of physical problems.

General expansion of natural power of linear combination of Bosonic operators in normal order