A systematic theory for studying certain types of
sequences of polynomials, or formal Laurent series,
through the use of modern algebra techniques.
The term
umbra
was coined by
J.J. Sylvester
in the mid
1800's, and originally referred to a symbol
used to represent a sequence of real numbers
.
Thus, if the sequences
,
and
satisfy
this could be written in
umbral notation
as
This notation is now obsolete, however.
The modern umbral calculus is designed to study
polynomial sequences
of binomial type,
that is, sequences for which
and
as well as
polynomial sequences
of Sheffer type,
that is, sequences for which
and
where
is a sequence of binomial type. Among the
class of Sheffer sequences are included sequences of
polynomials associated with the names of
Ch. Hermite,
E.N. Laguerre,
J. Bernoulli,
L. Euler,
S.D. Poisson
and
C. Charlier,
J. Meixner,
F.B. Pidduck,
S. Narumi,
G. Boole,
G.M. Mittag-Leffler,
F.W. Bessel,
E.T. Bell,
N.H. Abel,
and others.
If
is the algebra of polynomials in a single variable, then the dual space
is well-known to be a vector space. In fact,
is isomorphic to the vector space of formal power series
via the mapping
One may therefore identify
as the algebra
,
thus allowing for the multiplication of linear functionals. The algebra
is called the
umbral algebra.
In particular, for linear functionals
and
,
the geometric series
makes sense, and so one may define a sequence
of polynomials by
and the
orthogonality conditions
The sequences obtained in this way are precisely
the sequences of Sheffer type, and are called
Sheffer sequences.
The most powerful results in the umbral calculus come from a
study of the space of linear operators on the umbral algebra
.
If
is a linear operator on
,
its adjoint
is a linear operator on the umbral algebra
.
The most important linear operators on
are the
umbral operator,
defined for a sequence
of binomial type by
and the
umbral shift,
defined for a sequence
of binomial type by
Two key results in the umbral calculus say that a linear operator on
is an umbral operator if and only if its adjoint is an automorphism of
,
and an operator on
is an umbral shift if and only if its adjoint is a derivation on
.
The first result leads to an explicit formula for the polynomials
,
and the second result leads to a recurrence relation for the
,
which gives well-known recurrences in the case
of Hermite, Laguerre, Bernoulli, and other sequences.
Recently, the umbral calculus has been extended in several
directions. One direction is to the study of
non-Sheffer sequences, such as the sequences of
Chebyshev, Gegenbauer and Jacobi polynomials. Another direction is to the so-called
-umbral calculus,
where the polynomial coefficients are replaced by the
Gaussian coefficients.
The Gaussian coefficients, or
-binomial coefficients
,
are defined by
where the so-called
-shifted factorials
are defined by
Here
is seen either as a formal variable or as a complex variable of absolute value
.
Using these
-binomial
coefficients one has the
-binomial formula:
If
,
satisfy
,
then
Currently a whole theory is emerging involving
"q-versions of various classical objects" :
-special
functions,
-gamma-function,
quantum groups,
-integrals,
-orthogonal
polynomials,
-hypergeometric
series,
-Haar
measure, etc., complete with
-versions
of the various interrelations between all these
objects. Cf. also (the editorial comments to)
Special functions;
Quantum groups
and
[a9]–
[a10].
Finally, the umbral calculus has been generalized to
study sequences of formal Laurent series, where the logarithm plays a key role.