On Subnomials

Open access


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].

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics 1(1):41–46 1990.

  • [2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics 1(1):107–114 1990.

  • [3] Czesław Byliński. Some properties of restrictions of finite sequences. Formalized Mathematics 5(2):241–245 1996.

  • [4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics 1(1): 55–65 1990.

  • [5] 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.

  • [6] Jacek Gancarzewicz. Arytmetyka. Wydawnictwo UJ Kraków 2000. In Polish.

  • [7] 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.

  • [8] 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.

  • [9] Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics 6(4): 573–577 1997.

  • [10] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics 1(5):887–890 1990.

  • [11] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics 1(5):829–832 1990.

  • [12] W.W. Sawyer. The Search for Pattern. Penguin Books Ltd Harmondsworth Middlessex England 1970.

  • [13] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics 1(3):569–573 1990.

Journal information
Impact Factor

CiteScore 2018: 0.42

SCImago Journal Rank (SJR) 2018: 0.111
Source Normalized Impact per Paper (SNIP) 2018: 0.169

Target audience:

researchers in the fields of formal methods and computer-checked mathematics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 126 45 3
PDF Downloads 46 26 2