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## Gosper’s Algorithm

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**Gosper’s Algorithm**By Zachary Vogel**Binomial Coefficients**• Binomial Coefficients • The Binomial Theorem**Pascal’s Triangle**• Base identity for Pascal’s Triangle • 1 • 1 1 • 1 2 1 • 1 3 3 1 • 1 4 6 4 1 • 5 10 10 5 1 • 1 6 15 20 15 6 1**Binomial Identities**• Parallel Summation identity • Negation identity**Binomial Identities**• There are volumes of identities with binomial coefficients. • Here is one taken from a book: • Nanjundiah’s identity**Visual of Parallel Summation**1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 n=0**Visual of Parallel Summation**1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 n=1**Visual of Parallel Summation**1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 • 5 10 10 5 1 1 6 15 20 15 6 1 n=2**Visual of Parallel Summation**1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 • 5 10 10 5 1 1 6 15 20 15 6 1 n=3**Hypergeometrics**• How to find some order in all these identities with binomial coefficients? • Hypergeometric notation can be used to standardize identities.**Special Cases**• Exponential series • Geometric series**Hypergeometric Terms**• General Form of a hypergeometric term**Successive Terms**• If you take the ratio of successive terms of a hypergeometric series, the ratio is a rational polynomial of k. • If the ratio of successive terms form a rational function of k, then the series is hypergeometric up to a constant multiple.**Gosper’s Algorithm Overview**• Takes a hypergeometric term and sums it indefinitely • Example**Gosper’s Algorithm Overview**• The algorithm determines if the sum is a multiple of another hypergeometric term –OR – • It determines that the sum cannot be put in this form.**Gosper’s Algorithm Step 1**• We will assume that some such T(k) exists. • If we get an impossible situation, then no such T(k) exists. • The first step is to work out the term ratio of the summand t(k)**Term Ratio**• Write the term ratio for t(k) • where initially p(k) = 1 • We will pull out some factors of q and r into p.**Gosper’s Algorithm Step 1**• We require that q(k) and r(k) must have no factors • such that**Gosper’s Algorithm Step 1**• For if • then divide out the factors from q(k) and r(k) and absorb those terms into p(k) as follows • p(k+1)/p(k) telescopes nicely.**Gosper’s Algorithm Step 2**• Cleverly set • s(k) is some unknown function which will be the focus of the remainder of the algorithm • If we can determine what s(k) is, we can determine the final summation T(k).**Gosper’s Algorithm Step 2**• By applying • we get • We look to solve for s(k).**Unknown function s(k)**• In order to determine T(k) we must solve for s(k). This requires a few steps • Determine that s(k) is a rational function of k • Determine that s(k) is a polynomial in k • Determine a bound on the degree of s(k)**s(k) is a rational function of k**• By substitution • and • With the left hand side a rational function of k, and p(k) and r(k) are polynomials, s(k) must be a rational function of k**s(k) is a polynomial**• Knowing s(k) is a rational function of k, we can write it as the quotient of polynomials • such that f(k) and g(k) have no common factor • we will also assume that g(k) has a root, then find a contradiction • any polynomial without a root is just a constant, so s(k) will, itself, be a polynomial**s(k) is a polynomial**• Suppose that g(a) = g(b) = 0, and b-a is a nonnegative integer. (In particular, we might have a = b). • Since • We have**s(k) is a polynomial**• Substitute a=k+1, and separately b = k, we get: • Since f and g have no common root, • So either g(a-1)=0, or g(b+1)=0, or both r(a-1) = q(b) = 0. • The last choice is impossible by construction.**s(k) is a polynomial**• We now know that g(b+1) or g(a-1) is a root • By repeating this argument with a-1 and b, or a and b+1, we get infinitely many roots for g(k). • Therefore, g(k) has no root, thus is a constant. So s(k) is, itself a polynomial.**If we know a bound to the degree d of s(k), then we can**solve it by a system of d+1 linear equations, as given by the equation: Bounding degree of s(k)**By manipulating our previous equations, it can be seen that**With Bounding degree of s(k) We also Know change to ≤**Bounding degree of s(k)**• Now if Then the degree of the RHS will be Therefore, Otherwise, one of two options will occur i) ii) remove RHS**Solving s(k)**• Knowing the degree of s(k), solve • Then simply plug the known s(k) into**Example of Gosper’s Algorithm**• To provide an example of Gosper’s algorithm at work we will attempt to solve the negation identity • To begin we set our t(k) to the summand**Negation Identity**• Now by setting up a term ratio we will arrive at values for r(k), q(k) and p(k): • This satisfies the conditions on r(k) and q(k), so long as n is non-negative.**Negation Identity**• The next step is to determine s(k). • We can bound the degree by calculating R(k) and Q(k)**Negation Identity**• Since deg(R(k)) > deg(Q(k)), we have two options: d=0 or d=n. We will try d = 0 first. • So**Solution to negation identity**• Now we know our s(k) = -1/n, so we plug in to get T(k):**Solution to negation identity**• So Gosper’s algorithm gives the identity