Some Probabilistic Interpretations of the Multinomial Theorem

In my second Memoir “On the Calculus of Symbols,” I worked out the general case of multiplication according to one of the two systems of combination of non-commutative symbols previously given. In the present paper I propose to investigate the general case of multiplication according to the other system. I commence with the Binomial Theorem, to which the second system gives rise. In my previous researches I obtained the general term of the binomial theorem when the symbols combine according to the first system by equating symbolical coefficients; here, on the other hand, I consider the nature of the combinations which arise from the symbolical multiplication, and obtain the general term by summation. I next proceed to the multiplication of binomial factors. Here the general term is obtained by considering the alteration of weight undergone by certain symbols in the process of multiplication. The multinomial theorem according to the second system is next considered and its general term calculated. I conclude the memoir with some applications of the calculus of symbols to successive differentiation. This paper completes the investigation of symbolical multiplication and division according to the two systems of combination, the general case of division having been worked out by Mr. Spottiswoode in a very beautiful memoir recently published in the Transactions of this Society.

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XVII. On the calculus of symbols.—Third Memoir

Proceedings of the Royal Society of London ◽

10.1098/rspl.1862.0038 ◽

1863 ◽

Vol 12 ◽

pp. 184-184

Keyword(s):

General Term ◽

Binomial Theorem ◽

Multinomial Theorem

The following paper is a continuation of the two preceding Memoirs on the same subject. It has a fourfold object. In the first place, I calculate the general values of the coefficients in the Binomial Theorem given in the first Memoir. In the next place, I give an expression for the form of the coefficient of the general term of the multinomial theorem as previously explained. I then give a theorem for the multiplication of symbolical factors emanating from each other after a given law; and lastly, I investigate a binomial theorem, reciprocal to the binomial theorem already considered.

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