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Navigating the Identities List

            The identities list is large and only promises to get larger.  We offer this brief tutorial to get around this library under constant construction.  (Or is it discrete construction?)

            For all identities, we categorize an equation as (more difficult thing) = (less difficult thing).  We have five major categories of identities, categorized by the more difficult side of the equation.

Sums:  These involve the binomial coefficients (b.c.'s) being added together to form another number, or non-b.c.'s being added together to form a b.c.; there are no extra coefficients multiplying the b.c.'s, except for the occasional 2 in the case of row doubling identities, as in or perhaps an extra 1 thrown in, as in  .  If more multiplication takes place, as in  , this will be clasified a sum of products and be listed in that major category.

Products:  Quite simply, in these identities, the addition sign is only found in the numerators and denominators of fractions or in the upper and lower indices of B.C.'s; an exception is made for Cat. #2500001, which is a sum of products, but it is the generator of the Catalan numbers, Cat. #2500002, which is fairly classified as a product, .

Sums of products:  This is a large category, and the products are split up into three main sub-categories: purely B.C. products, alternating products (where only a (-1)^k is included with the B.C.'s) and mixed products, where any other coefficients may be included with the B.C.s.  Beyond that, sub-classifications involve whether the identities sums to a B.C., a sum or product of B.C.s or a non-B.C.

Products of sums:  This is currently a sparsely populated category, but that may change in the future; when the number of identities in this category gets over a handful, we will give navigation advice.

Factorization identities:  Almost all the identities here deal with mod p or mod pⁿ, where p is a prime.

            Beyond these categories, we also have a list of the identities that have been named; somewhat idiosyncratically, the names listed as Pascal's n^th Identity are the names used by the writer of this website only; in the webpage explaining Pascal's work, the writer renumbered Pascal's original 19 corollaries to combine into a single identity two identities that are mirror symmetries of one another.

            Also added for the researcher is a list of identities that involve "famous" numbers; besides listing all the identities that use the Euler and Stirling numbers, we also have a list of those identities that sum to powers of 2 and sum almost always to zero, which includes any Kronecker delta solution.

            If you have any questions about getting around the identity page, please feel free to contact us at the e-mail address mhubbard(at)bay(dot)csuhayward(dot)edu.