By Wolfram Koepf
ISBN-10: 3528069503
ISBN-13: 9783528069506
During this e-book, glossy algorithmic thoughts for summation--most of that have been brought in the final decade--are constructed and thoroughly applied through machine algebra procedure software program (which will be downloaded from the net; URL is given within the text). The algorithms of Gosper, Zeilberger, and Petkovsek on hypergeometric summation and recurrence equations and their $q$-analogues are lined, and comparable algorithms on differential equations are thought of. An similar concept of hyperexponential integration as a result of Almkvist and Zeilberger completes the quantity. the mix of all effects thought of provides paintings with orthogonal polynomials and (hypergeometric kind) designated features a fantastic algorithmic starting place. for that reason, many examples from this very lively box are given. The ebook is designed to be used as framework for a seminar at the subject, yet can also be appropriate to be used in a complicated lecture direction.
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Extra info for Hypergeometric summation. An algorithmic approach to summation and special function identities
Sample text
This result of course coincides with that obtained in the first solution. It shows that among the numbers under consideration there are more with I's among their digits than without. = - 52 SOLUTIONS llb. Among the integers from I to 222,222,222 there are 22,222,222 ending in a 0 (namely, the numbers 10,20,30, ... , 222,222,220). In order to determine how many integers have a 0 in the next to last position, notice that what comes before this 0 can be anything from I to 2,222,222, while what comes after it can be anything from 0 to 9.
Then g is counted by each term of (I), and is therefore counted a net of I + I + I - I - I - I + I = I time. This analysis shows that expression (I) counts each element of A V B V C once. On the other hand, elements not in A V B V C are not counted in any of its terms, and therefore (I) is equal to #(A V B V C). 12c. The general case can be treated by the same reasoning as that used in part b. We must show that in the expression #(A 1) + #(AJ + ... + #(Am) - #(Al (\ AJ - #(Al (\ Aa) - ... - #(A mAm) + #(Al (\ A2 (\ Aa) + ...
Suppose that as N -- 00 the ratio q(N)IN approaches a limit; in this case this limit is called the probability that a number selected at random from the entire sequence has the desired property. Note that this probability depends on the way in which the numbers are arranged in a sequence. Changing the order of the numbers can change the value of the probability. Example: consider the positive integers arranged in increasing order: 1,2,3, ... Of the first N of these numbers, [N/2] are even; as N -- 00 the ratio [NI2]/N approaches i, which means that the probability that any number selected at random is even equals 1/2.
Hypergeometric summation. An algorithmic approach to summation and special function identities by Wolfram Koepf
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