generating function
Sequence-to-polynomial map
A generating function is a mathematical tool that encodes an infinite sequence of numbers as coefficients of a power series, allowing the sequence to be manipulated algebraically.
If you have a sequence like 1, 1, 2, 3, 5, 8... (the Fibonacci numbers), the generating function packages this into a single expression where each term's coefficient corresponds to an element in the sequence.
Generating functions are widely used in combinatorics, probability theory, and analysis to solve recurrence relations, count objects, and derive closed-form expressions for sequences.
Number multiplying a variable
- enumerationSequential listing of items
- ordinaryStandard, unremarkable
- exponentialIncreasing by constant ratio
- recurrenceReturn of something previous
convolution · extraction · partitions · permutations · combinations · binomial · labeled · unlabeled
Formula where terms depend on earlier ones
- arithmeticMathematics of basic operations
- geometricRelating to mathematical shapes
- harmonicOf musical overtones
- infinite seriesSum with unlimited terms
Process of approaching sameness
- singularityPoint where values explode
- poleCylindrical shaft or extremity
- residueMatter left behind
- divergenceSeparation from common origin
coefficient
Number multiplying a variable
- enumerationSequential listing of items
- ordinaryStandard, unremarkable
- exponentialIncreasing by constant ratio
recurrence · convolution · extraction · partitions · permutations · combinations · binomial · labeled · unlabeled · species
recurrence relation
Formula where terms depend on earlier ones
- arithmeticMathematics of basic operations
- geometricRelating to mathematical shapes
- harmonicOf musical overtones
convergence
Process of approaching sameness
- singularityPoint where values explode
- poleCylindrical shaft or extremity
- residueMatter left behind
discrete
Individually separate, unconnected
- continuousWithout pause or gap
- combinatoricspermutation, combinations, combination, counting, arrangements
- discrete mathcombinatorics, graph theory, set theory, number theory, proofs
- branching processes
- recursionrecursion, base case, self-reference, infinite recursion, recurrence relation
- special functions
- symmetric polynomials
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