Binomial transform

The binomial transform is a discrete transformation in mathematics that maps a sequence of numbers \{a_n\}_{n=0}^\infty to a new sequence \{b_n\}_{n=0}^\infty defined by the formula

b_n = \sum_{k=0}^n \binom{n}{k} a_k

for each nonnegative integer n, where \binom{n}{k} denotes the binomial coefficient.^[1] This operation can be viewed as a convolution of the original sequence with the sequence of binomial coefficients and is closely related to the computation of forward differences in discrete calculus.^[1] The transform is linear and bijective, meaning it has a well-defined inverse given by

a_n = \sum_{k=0}^n \binom{n}{k} (-1)^{n-k} b_k,

which incorporates alternating signs and preserves the structure of sequences over the nonnegative integers.^[1] In matrix terms, it corresponds to multiplication by an infinite lower triangular matrix whose entries are binomial coefficients, facilitating its representation in computational and algebraic contexts.^[1] Notable properties include its connection to exponential generating functions: if A(x) = \sum_{n=0}^\infty a_n \frac{x^n}{n!} is the exponential generating function of \{a_n\}, then the generating function of \{b_n\} is e^x A(x).^[2] This relationship underscores its utility in enumerative combinatorics, where it aids in deriving identities for special sequences such as powers, factorials, and harmonic numbers—for instance, the transform of the constant sequence a_n = 1 yields b_n = 2^n.^[1] Applications span combinatorics and analysis, including the acceleration of convergent series, the simplification of summation formulas, and the generation of identities involving Fibonacci numbers, Stirling numbers, and Laguerre polynomials.^[3] A key insight is its ability to convert multiplication by the index n in a sequence into applications of the backward difference operator, enabling elegant proofs of discrete analogs to continuous calculus results.^[3] These features make the binomial transform a fundamental tool in studying sequence behaviors and probabilistic structures like skip lists.^[4]

Fundamentals

Definition

The binomial transform is a sequence transformation that maps a given sequence \{a_n\}_{n=0}^\infty of complex numbers (or more generally, elements from any ring) to another sequence \{b_n\}_{n=0}^\infty via the formula

b_n = \sum_{k=0}^n \binom{n}{k} a_k,

where \binom{n}{k} denotes the binomial coefficient. The inverse binomial transform recovers the original sequence from the transformed one using

a_n = \sum_{k=0}^n (-1)^{n-k} \binom{n}{k} b_k. This inversion formula demonstrates that the transform is invertible, establishing it as a bijection on the space of all sequences. As a linear operator on the vector space of sequences (over the reals or complexes), the binomial transform acts by convolving the input sequence with the sequence of binomial coefficients $\{ \binom{n}{k} \}_{0 \leq k \leq n}$, preserving linearity: if $\{c_n\}$ and $\{d_n\}$ are sequences and $\alpha, \beta$ are scalars, then the transform of $\{\alpha c_n + \beta d_n\}$ is $\alpha$ times the transform of $\{c_n\}$ plus $\beta$ times the transform of $\{d_n\}$. Its bijectivity extends to the ring of formal power series, where the transform corresponds to multiplication by the generating function $(1 - x)^{-1}$, which is invertible in that ring. ### Examples One illustrative example of the binomial transform involves the constant sequence $ a_n = 1 $ for all $ n \geq 0 $. The transformed sequence is given by $ b_n = \sum_{k=0}^n \binom{n}{k} \cdot 1 = 2^n $, which counts the total number of subsets of an $ n $-element set.[](https://www.unk.edu/academics/math/_files/binomial-transformation1.pdf) Another basic case is the linear sequence $ a_n = n $. Here, the binomial transform yields $ b_n = \sum_{k=0}^n \binom{n}{k} k = n \cdot 2^{n-1} $, a result derived from differentiating the binomial theorem and evaluating at unity.[](https://mathworld.wolfram.com/BinomialSums.html) For the factorial sequence $ a_n = n! $, the transform produces $ b_n = \sum_{k=0}^n \binom{n}{k} k! $, which enumerates the number of injections from a set of size $ k $ to $ n $ summed over $ k $. To demonstrate patterns, the following table shows the binomial transforms for small values of $ n $ (up to 5) applied to the constant sequence ($ a_n = 1 $), the linear sequence ($ a_n = n $), the factorial sequence ($ a_n = n! $), and an exponential sequence ($ a_n = 2^n $, whose transform is $ b_n = 3^n $): | $ n $ | $ a_n = 1 $ | $ b_n $ | $ a_n = n $ | $ b_n $ | $ a_n = n! $ | $ b_n $ | $ a_n = 2^n $ | $ b_n = 3^n $ | |---------|---------------|-----------|----------------|-----------|----------------|-----------|------------------|-----------------| | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 3 | | 2 | 1 | 4 | 2 | 4 | 2 | 5 | 4 | 9 | | 3 | 1 | 8 | 3 | 12 | 6 | 16 | 8 | 27 | | 4 | 1 | 16 | 4 | 32 | 24 | 65 | 16 | 81 | | 5 | 1 | 32 | 5 | 80 | 120 | 326 | 32 | 243 | These computations highlight how the transform amplifies growth rates, turning polynomials into higher-degree forms scaled by powers of 2 and exponentials into shifted bases.[](https://mathworld.wolfram.com/BinomialSums.html)[](https://oeis.org/A000079) Applying the inverse binomial transform to the sequence $ b_n = 2^n $ recovers the original constant sequence $ a_n = 1 $. ## Generating Functions ### Ordinary Generating Functions The ordinary generating function of the binomial transform of a sequence relates to that of the original sequence through multiplication by $ \frac{1}{1-x} $ and composition with $ \frac{x}{1-x} $. Let $ A(x) = \sum_{n=0}^{\infty} a_n x^n $ denote the ordinary generating function of the sequence $ \{a_n\}_{n \geq 0} $. The binomial transform is defined by $ b_n = \sum_{k=0}^n \binom{n}{k} a_k $ for $ n \geq 0 $, with $ b_0 = a_0 $. The ordinary generating function of $ \{b_n\} $ is then \[ B(x) = \sum_{n=0}^{\infty} b_n x^n = \frac{1}{1-x} \, A\left( \frac{x}{1-x} \right).

This relation holds formally as power series and analytically in regions where both series converge.^[5] To derive this, substitute the definition of b_n into B(x):

B(x) = \sum_{n=0}^{\infty} \left( \sum_{k=0}^n \binom{n}{k} a_k \right) x^n = \sum_{k=0}^{\infty} a_k \sum_{n=k}^{\infty} \binom{n}{k} x^n,

where the double sum is interchanged under absolute convergence for |x| < 1. The inner generating function is the known series

\sum_{n=k}^{\infty} \binom{n}{k} x^n = \frac{x^k}{(1-x)^{k+1}},

arising from the negative binomial expansion (1-x)^{-(k+1)} = \sum_{m=0}^{\infty} \binom{m+k}{k} x^m after the substitution m = n - k. Thus,

B(x) = \sum_{k=0}^{\infty} a_k \frac{x^k}{(1-x)^{k+1}} = \frac{1}{1-x} \sum_{k=0}^{\infty} a_k \left( \frac{x}{1-x} \right)^k = \frac{1}{1-x} \, A\left( \frac{x}{1-x} \right).

The inverse relation recovers the original generating function from the transform. The inverse binomial transform is given by a_n = \sum_{k=0}^n (-1)^{n-k} \binom{n}{k} b_k, and solving the forward relation yields

A(x) = \frac{1}{1+x} \, B\left( \frac{x}{1+x} \right).

This follows by substituting y = x / (1 + x) into the forward formula adjusted for the sign alternation, confirming invertibility.^[6] In analytic contexts, convergence of B(x) requires |x| < 1 and \left| \frac{x}{1-x} \right| within the radius of convergence of A(x); similarly for the inverse. The transformation preserves formal power series operations but may alter the radius of convergence depending on the specific A(x).

Exponential Generating Functions

The exponential generating function (EGF) for a sequence \{a_n\}_{n \geq 0} is defined as A(x) = \sum_{n=0}^\infty a_n \frac{x^n}{n!}. For the binomial transform \{b_n\} of \{a_n\}, where b_n = \sum_{k=0}^n \binom{n}{k} a_k, the corresponding EGF B(x) = \sum_{n=0}^\infty b_n \frac{x^n}{n!} satisfies B(x) = e^x A(x).^[7]^[2] To derive this relation, consider the coefficient of x^n / n! in B(x). By definition, \frac{b_n}{n!} = \sum_{k=0}^n \frac{a_k}{k!} \cdot \frac{\binom{n}{k} k!}{n!}. Simplifying the binomial factor gives \binom{n}{k} \frac{k!}{n!} = \frac{1}{(n-k)!}, so \frac{b_n}{n!} = \sum_{k=0}^n \frac{a_k}{k!} \cdot \frac{1}{(n-k)!}. This is precisely the coefficient of x^n / n! in the product A(x) e^x, since e^x = \sum_{m=0}^\infty \frac{x^m}{m!}. Thus, B(x) = e^x A(x).^[7] The transformation is invertible, with the inverse binomial transform recovering the original sequence via A(x) = e^{-x} B(x). This follows directly from multiplying both sides of the forward relation by e^{-x}. For sequences involving factorials, such as when a_n = \tilde{a}_n \cdot n! where \{\tilde{a}_n\} has ordinary generating function \tilde{A}(x), the EGF A(x) = \tilde{A}(x) simplifies the binomial transform to an ordinary generating function multiplication by e^x, relating it to the Borel transform.^[2]

Inverse Binomial Transform

The inverse binomial transform recovers the original sequence \{a_n\}_{n=0}^\infty from the transformed sequence \{b_n\}_{n=0}^\infty, where the forward transform is defined as b_n = \sum_{k=0}^n \binom{n}{k} a_k. The explicit formula for the inverse is given by

a_n = \sum_{k=0}^n (-1)^{n-k} \binom{n}{k} b_k

for each n \geq 0https://www.unk.edu/academics/math/_files/binomial-transformation1.pdf^[8]. In matrix form, the forward binomial transform corresponds to an infinite lower triangular matrix B with entries B_{n,k} = \binom{n}{k} for $0 \leq k \leq n and zero otherwise, such that the vector \mathbf{b} = B \mathbf{a}https://www.unk.edu/academics/math/_files/binomial-transformation1.pdf. The inverse matrix B^{-1} is also lower triangular, with entries B^{-1}_{n,k} = (-1)^{n-k} \binom{n}{k} for $0 \leq k \leq n and zero otherwise, ensuring \mathbf{a} = B^{-1} \mathbf{b}https://www.unk.edu/academics/math/_files/binomial-transformation1.pdf. The invertibility of the binomial transform follows from the composition property in the context of ordinary generating functions. Let A(x) = \sum_{n=0}^\infty a_n x^n and B(x) = \sum_{n=0}^\infty b_n x^n. The forward transform yields B(x) = \frac{A\left(\frac{x}{1-x}\right)}{1-x}, while the inverse yields A(x) = (1+x)^{-1} B\left(\frac{x}{1+x}\right). Composing the forward and inverse operations results in the identity, as the generalized binomial transform satisfies B(1,1) \circ B(-1,1) = B(0,1), where B(0,1) is the identity operator. Computationally, the inverse can be evaluated directly using the summation formula, which requires O(n^2) arithmetic operations for the first n+1 terms due to the triangular structure of the matrix. For larger n, faster methods leverage the generating function relations, such as series expansion or convolution via fast Fourier transform on the coefficients, achieving O(n \log n) complexity in suitable implementations.

Euler Transform

The Euler transform of a sequence \{a_n\}_{n=0}^\infty is defined by

e_n = \sum_{k=0}^n (-1)^k \binom{n}{k} a_k

for n \geq 0.^[9] This operation introduces an alternating sign in the summation, distinguishing it from the unsigned binomial transform. The Euler transform can be understood as the standard binomial transform applied to the modified sequence \{(-1)^k a_k\}_{k=0}^\infty.^[10] A primary application of the Euler transform lies in accelerating the convergence of alternating series of the form S = \sum_{n=0}^\infty (-1)^n a_n, where \{a_n\} consists of positive terms with a_n nondecreasing initially. The transformed series is then given by

S = \sum_{n=0}^\infty \frac{e_n}{2^{n+1}},

which typically converges more rapidly than the original, especially for slowly converging cases.^[9] For instance, applying the Euler transform to the series for e^{-1} = \sum_{n=0}^\infty (-1)^n / n! (with a_n = 1/n!) yields a new series that converges faster by reweighting terms via the binomial coefficients and signs.^[10] The Euler transform is named after Leonhard Euler, who developed foundational methods for series acceleration, including transformations for alternating and divergent series, during the 18th century.^[11] The ordinary generating function for the Euler-transformed sequence \{e_n\} takes the form E(x) = \frac{1}{1 - x} f\left( -\frac{x}{1 - x} \right), where f(x) is the generating function for \{a_n\}.

Properties and Representations

Binomial Convolution

The binomial transform of a sequence a = (a_n)_{n \geq 0} admits an interpretation as a convolution operation. Specifically, it corresponds to the binomial convolution of a with the constant sequence u = (u_n)_{n \geq 0} where u_k = 1 for all k \geq 0, yielding b_n = (a \star u)_n = \sum_{k=0}^n \binom{n}{k} a_k u_{n-k} = \sum_{k=0}^n \binom{n}{k} a_k.^[12] The binomial convolution \star equips the set of all sequences over a commutative ring with identity with an algebraic structure. This set forms a ring under \star, where addition is componentwise and multiplication is given by the convolution.^[13] The identity element for this ring is the Kronecker delta sequence \delta = (\delta_n)_{n \geq 0} with \delta_0 = 1 and \delta_n = 0 for n \geq 1, since (a \star \delta)_n = \sum_{k=0}^n \binom{n}{k} a_k \delta_{n-k} = a_n.^[13] The operation inherits key properties from its convolutional nature. Binomial convolution is commutative, as \sum_{k=0}^n \binom{n}{k} a_k b_{n-k} = \sum_{k=0}^n \binom{n}{k} b_k a_{n-k}, and associative, reflecting the ring structure.^[13]^[12] In umbral calculus, binomial convolution underpins the theory of polynomial sequences of binomial type, which are precisely those closed under this operation in the sense that p_n(x + y) = \sum_{k=0}^n \binom{n}{k} p_k(x) p_{n-k}(y).^[14] This closure enables operator interpretations, such as representing the falling factorial basis via powers of the forward difference operator \Delta = e^D - 1, where D is the differentiation operator and \Delta^n f(x) = \sum_{k=0}^n \binom{n}{k} (-1)^{n-k} f(x + k).^[14]

Integral Representation

A key integral representation involves the Beta function, which provides an exact expression for the reciprocal of the binomial coefficient appearing in the transform kernel: \int_0^1 (1 - x)^{n-k} x^k \, dx = B(n - k + 1, k + 1) = \frac{1}{(n+1) \binom{n}{k}}. The derivation follows from the integral definition of the Beta function B(\alpha, \beta) = \int_0^1 x^{\alpha-1} (1-x)^{\beta-1} \, dx = \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)}, specialized to integer arguments where \Gamma(m+1) = m!.^[15] In probability theory, these integral representations underpin the beta-binomial distribution, a compound distribution where the success probability in a binomial model is integrated over a Beta prior. The probability mass function is P(K = k \mid n, \alpha, \beta) = \binom{n}{k} \frac{B(k + \alpha, n - k + \beta)}{B(\alpha, \beta)}, which explicitly incorporates the Beta integral to normalize the mixture, relating moments of the binomial distribution to expectations under the continuous Beta measure. This connection highlights how the binomial transform's kernel, via its inverse Beta form, models overdispersion in count data beyond the standard binomial. The integral forms also enable asymptotic analysis for large n, where approximations to the Beta function yield insights into the behavior of the binomial transform. For instance, when parameters are large, Stirling's approximation or series expansions of \log B(p, q) provide high-order estimates, such as B(p, q) \sim \sqrt{2\pi} \, p^{p-1/2} q^{q-1/2} (p+q)^{-(p+q-1/2)} \exp\left( \frac{1}{12} \left( \frac{1}{p} + \frac{1}{q} - \frac{1}{p+q} \right) \right) for fixed p/q ratio, allowing large-n limits of transformed sequences like central binomial sums. These methods, akin to Laplace's approximation on the integral, quantify the scale of b_n relative to $2^n times sequence averages.^[16]

Generalizations and Extensions

Multivariate Binomial Transform

The multivariate binomial transform generalizes the univariate binomial transform to sequences indexed by multi-indices \mathbf{n} = (n_1, \dots, n_d) in d dimensions, where each n_i \ge 0 is an integer. For a sequence \{a_{\mathbf{k}}\} with \mathbf{k} = (k_1, \dots, k_d), the transformed sequence \{b_{\mathbf{n}}\} is defined by the multidimensional convolution

b_{\mathbf{n}} = \sum_{k_1=0}^{n_1} \cdots \sum_{k_d=0}^{n_d} \left( \prod_{i=1}^d \binom{n_i}{k_i} \right) a_{\mathbf{k}}.

^[17] This form reflects the separable nature of the transform across dimensions, where the product of binomial coefficients arises from applying the univariate transform independently to each coordinate. The multivariate ordinary generating function for the transformed sequence is

B(x_1, \dots, x_d) = \sum_{\mathbf{n}} b_{\mathbf{n}} x_1^{n_1} \cdots x_d^{n_d} = \frac{A(x_1, \dots, x_d)}{\prod_{i=1}^d (1 - x_i)},

where A(x_1, \dots, x_d) is the generating function for \{a_{\mathbf{k}}\}. This relation follows from the product rule for multivariate generating functions and the univariate identity B(x) = A(x)/(1 - x). In multivariate statistics, the transform finds applications in analyzing moments of distributions with independent components, such as the joint moments of a multinomial distribution, where the product structure simplifies computations of higher-order statistics like cumulants or factorial moments. The transform is invertible, with the inverse given by a signed multidimensional sum

a_{\mathbf{n}} = \sum_{k_1=0}^{n_1} \cdots \sum_{k_d=0}^{n_d} \left( \prod_{i=1}^d \binom{n_i}{k_i} (-1)^{n_i - k_i} \right) b_{\mathbf{k}}.

^[17] This corresponds to the action of the lower triangular multidimensional binomial matrix with alternating signs on the diagonal blocks, ensuring bijectivity analogous to the univariate case.

q-Binomial Transform

The q-binomial transform is a q-analogue of the classical binomial transform, defined for a sequence \{a_k\}_{k=0}^\infty by the sequence \{b_n\}_{n=0}^\infty where

b_n = \sum_{k=0}^n \binom{n}{k}_q a_k,

^[18] and the q-binomial coefficient \binom{n}{k}_q is given by

\binom{n}{k}_q = \prod_{i=1}^k \frac{1 - q^{n - k + i}}{1 - q^i}

for integers $0 \leq k \leq n and |q| < 1. This coefficient counts the number of k-dimensional subspaces of an n-dimensional vector space over the finite field \mathbb{F}_q, providing a combinatorial interpretation, and satisfies the q-Pascal identity \binom{n}{k}_q = q^k \binom{n-1}{k}_q + \binom{n-1}{k-1}_q. As q \to 1, the q-binomial coefficient reduces to the ordinary binomial coefficient, recovering the standard binomial transform. Key properties include q-invertibility, where the inverse transform recovers the original sequence via a_n = \sum_{k=0}^n (-1)^{n-k} q^{(n-k)(n-k-1)/2} \binom{n}{k}_q b_k, analogous to the signed inverse in the classical case but deformed by quadratic q-exponents.^[19] This invertibility follows from the generating function relation and the q-analogue of the binomial inversion formula. Additionally, the transform connects to quantum groups, as q-binomial coefficients parameterize representations of U_q(sl_2) and appear in quantum symmetric functions, facilitating studies in deformed algebraic structures. Applications of the q-binomial transform abound in q-series identities and partition theory, where it aids in deriving summation formulas for basic hypergeometric functions, such as transformations between {}_2\phi_1 series. In partition theory, it underpins q-analogues of classical identities, including the q-Euler partition theorem, which equates the generating function for partitions into distinct parts to that for odd parts via products involving q-binomials. These tools are essential for proving Rogers-Ramanujan identities and analyzing weighted partition statistics in basic hypergeometric contexts.