Sequences of Binomial Type
by aoum, Apr 4, 2025, 10:37 PM
Sequences of Binomial Type
In the language of combinatorics and algebra, a sequence of binomial type is a polynomial sequence that generalizes the behavior of the powers
under the binomial expansion. These sequences naturally arise in the study of combinatorial identities, generating functions, and umbral calculus.
1. Definition
Let
be a sequence of polynomials, each of degree 
. The sequence is said to be of binomial type if it satisfies the identity:
This is a natural generalization of the classical binomial theorem:
2. Examples of Binomial Type Sequences
3. Generating Function Characterization
A sequence
is of binomial type if and only if it has an exponential generating function of the form:
where
is a formal power series with 
and 
. For example:
4. Properties
Let be a sequence of binomial type. Then:
5. Umbral Calculus Interpretation
In umbral calculus, one introduces a linear functional
such that 
for an abstract umbra 
. Then the binomial identity becomes:
![$$
\mathcal{L}[(a + b)^n] = \sum_{k=0}^n \binom{n}{k} \mathcal{L}(a^k) \mathcal{L}(b^{n-k})
$$](//latex.artofproblemsolving.com/e/e/4/ee46118b3911fe932b900aa5e5720cb4353d5b9e.png)
This is a symbolic way of writing
in terms of 
and 
.
6. Sheffer Classification
Every sequence of binomial type is a special case of a Sheffer sequence. A Sheffer sequence
has a generating function:
If
, then the sequence is of binomial type.
7. Orthogonality and Combinatorics
Sequences of binomial type appear in numerous combinatorial contexts:
8. Proof of Binomial Identity for
Let’s verify the binomial identity for the monomial sequence:
This follows directly from the classical binomial theorem. Hence,
is of binomial type.
9. Proof That Falling Factorials Are of Binomial Type
We claim that:
This follows from the combinatorics of choosing
elements to place in 
and the remaining in 
in orderings.
10. Conclusion
Sequences of binomial type generalize the binomial theorem to a wide class of polynomial sequences. Their combinatorial and algebraic structure plays a central role in generating function theory, umbral calculus, and discrete mathematics. They provide a unifying framework for many familiar polynomial families and are foundational in symbolic combinatorics.
References
In the language of combinatorics and algebra, a sequence of binomial type is a polynomial sequence that generalizes the behavior of the powers
under the binomial expansion. These sequences naturally arise in the study of combinatorial identities, generating functions, and umbral calculus.1. Definition
Let
be a sequence of polynomials, each of degree 
. The sequence is said to be of binomial type if it satisfies the identity:
This is a natural generalization of the classical binomial theorem:
2. Examples of Binomial Type Sequences
- Monomials:
- Falling Factorials:
- Abel Polynomials:
for fixed 
- Touchard Polynomials (or Bell polynomials): arising in the context of partitioning sets
- Binomial Convolution Polynomials: arising from exponential generating functions
for fixed 
3. Generating Function Characterization
A sequence
is of binomial type if and only if it has an exponential generating function of the form:
where
is a formal power series with 
and 
. For example:- For , we have
, since 
- For
, the generating function is 
, so 
,
,
,
4. Properties
Let
be a sequence of binomial type. Then:
for 
satisfies a convolution identity:
- The sequence is closed under differentiation: there exists a sequence
such that
if and only if the sequence is also an Appell sequence, i.e.,
for 
satisfies a convolution identity:
such that
5. Umbral Calculus Interpretation
In umbral calculus, one introduces a linear functional
such that 
for an abstract umbra 
. Then the binomial identity becomes:![$$
\mathcal{L}[(a + b)^n] = \sum_{k=0}^n \binom{n}{k} \mathcal{L}(a^k) \mathcal{L}(b^{n-k})
$$](http://latex.artofproblemsolving.com/e/e/4/ee46118b3911fe932b900aa5e5720cb4353d5b9e.png)
This is a symbolic way of writing
in terms of 
and 
.6. Sheffer Classification
Every sequence of binomial type is a special case of a Sheffer sequence. A Sheffer sequence
has a generating function:
If
, then the sequence is of binomial type.7. Orthogonality and Combinatorics
Sequences of binomial type appear in numerous combinatorial contexts:
- Enumerating labeled structures via exponential generating functions
- Bell numbers and set partitions
- Combinatorial identities involving Stirling and Lah numbers
- Polynomial interpolation, recurrence relations, and difference calculus
8. Proof of Binomial Identity for
Let’s verify the binomial identity for the monomial sequence:
This follows directly from the classical binomial theorem. Hence,
is of binomial type.9. Proof That Falling Factorials Are of Binomial Type
We claim that:
This follows from the combinatorics of choosing
elements to place in 
and the remaining in 
in orderings.10. Conclusion
Sequences of binomial type generalize the binomial theorem to a wide class of polynomial sequences. Their combinatorial and algebraic structure plays a central role in generating function theory, umbral calculus, and discrete mathematics. They provide a unifying framework for many familiar polynomial families and are foundational in symbolic combinatorics.
References
- Roman, S. The Umbral Calculus, Dover, 2005
- Rota, G.-C. Finite Operator Calculus
- Wikipedia: Polynomial Sequences of Binomial Type
April 2025
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