On Subnomials

While discussing the sum of consecutive powers as a result of division of two binomials W.W. Sawyer [12] observes “It is a curious fact that most algebra textbooks give our ast result twice. It appears in two different chapters and usually there is no mention in either of these that it also occurs in the other. The first chapter, of course, is that on factors. The second is that on geometrical progressions. Geometrical progressions are involved in nearly all financial questions involving compound interest – mortgages, annuities, etc.” It’s worth noticing that the first issue involves a simple arithmetical division of 99...9 by 9. While the above notion seems not have changed over the last 50 years, it reflects only a special case of a broader class of problems involving two variables. It seems strange, that while binomial formula is discussed and studied widely [7], [8], little research is done on its counterpart with all coefficients equal to one, which we will call here the subnomial. The study focuses on its basic properties and applies it to some simple problems usually proven by induction [6].Department of Carbohydrate Technology, University of Agriculture, Krakow, PolandGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Czesław Byliński. Some properties of restrictions of finite sequences. Formalized Mathematics, 5(2):241–245, 1996.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Wenpai Chang, Hiroshi Yamazaki, and Yatsuka Nakamura. The inner product and conjugate of finite sequences of complex numbers. Formalized Mathematics, 13(3):367–373, 2005.Jacek Gancarzewicz. Arytmetyka. Wydawnictwo UJ, Kraków, 2000. In Polish.Cao Hui-Qin and Pan Hao. Factors of alternating binomial sums. Advances in Applied Mathematics, 45(1):96 – 107, 2010. doi:http://dx.doi.org/10.1016/j.aam.2009.09.004.Sudesh K. Khanduja, Ramneek Khassa, and Shanta Laishram. Some irreducibility results for truncated binomial expansions. Journal of Number Theory, 131(2):300 – 308, 2011. doi:http://dx.doi.org/10.1016/j.jnt.2010.08.004.Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4): 573–577, 1997.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829–832, 1990.W.W. Sawyer. The Search for Pattern. Penguin Books Ltd, Harmondsworth, Middlessex, England, 1970.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990