DiscreteMath`RSolve`

A recurrence or difference equation specifies a relationship between different values of an unknown sequence. For example, the equation a[n] a[n-1] + a[n-2] specifies that in the unknown sequence a[n], the sum of the two previous terms gives the next term.

Mathematica includes the built-in function RSolve to find solutions to equations of this type. The RSolve.m package contains an assortment of utility functions useful for investigating methods of solving recurrence equations.

Generating functions of sequences.

The function is called the generating function of the solution . The exponential generating function of the sequence is the function . For example, is the generating function of the sequence and the exponential generating function of the constant sequence .

This loads the package.

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This computes , which is simply a geometric series.

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This computes the generating function of the sequence of squares of integers.

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This confirms that the coefficients of the series expansion of the function are the squares of the integers.

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This exponential generating function is simply the Taylor series for the exponential.

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Here is the exponential generating function of the sequence of squares of integers.

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This computes the exponential power sum for a series that has a different form when is odd and even. The function Even is provided in the package. It is equivalent to the built-in EvenQ, except that it does not evaluate when its argument is symbolic.

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Generating functions for the solutions to recurrence equations.

The solution to the given recurrence equation is the sequence of Fibonacci numbers. Thus, this gives the generating function for this sequence.

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This recurrence equation gives the sequence of Bernoulli numbers.

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This computes a list of coefficients in the power series of the generating function just computed and then cancels a factor of from the term.

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The resulting list contains Bernoulli numbers.

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Coefficients of power series.

This gives the coefficient of in the power series expansion of about .

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The coefficient of a series can be computed for a general value of . The default is to assume that .

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Here no assumption is made about .

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Here is another general coefficient where it is assumed that .

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When no assumption is made about , the result involves the conditional If.

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The options GeneratingFunctionConstants and ExponentialGeneratingFunctionConstants can be used to specify constants in the GeneratingFunction and ExponentialGeneratingFunction solutions, respectively.

Options for GeneratingFunction and ExponentialGeneratingFunction.