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In particular, for linear functionals $L$ and $M$, the geometric series $M,ML,ML^2,\ldots$ makes sense, and so one may define a sequence $s_n(x)$ of polynomials by $\deg s_n(x) = n$ and the orthogonality conditions
 
In particular, for linear functionals $L$ and $M$, the geometric series $M,ML,ML^2,\ldots$ makes sense, and so one may define a sequence $s_n(x)$ of polynomials by $\deg s_n(x) = n$ and the orthogonality conditions
 
$$
 
$$
ML^k(s_n(x)) = \delta_{kn} \ .
+
ML^k(s_n(x)) = n! \delta_{kn} \ .
 
$$
 
$$
 
The sequences obtained in this way are precisely the sequences of Sheffer type, and are called Sheffer sequences.
 
The sequences obtained in this way are precisely the sequences of Sheffer type, and are called Sheffer sequences.
Line 36: Line 36:
 
The most powerful results in the umbral calculus come from a study of the space of linear operators on the umbral algebra $\mathcal{A}$. If $T : P \rightarrow P$ is a linear operator on $P$, its adjoint $T^* : \mathcal{A} \rightarrow \mathcal{A}$ is a linear operator on the umbral algebra $\mathcal{A}$.
 
The most powerful results in the umbral calculus come from a study of the space of linear operators on the umbral algebra $\mathcal{A}$. If $T : P \rightarrow P$ is a linear operator on $P$, its adjoint $T^* : \mathcal{A} \rightarrow \mathcal{A}$ is a linear operator on the umbral algebra $\mathcal{A}$.
  
The most important linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505034.png" /> are the umbral operator, defined for a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505035.png" /> of binomial type by
+
The most important linear operators on $P$ are the ''umbral operator'', defined for a sequence $p_n(x)$ of binomial type by
 +
$$
 +
\lambda(x^n) = p_n(x)
 +
$$
 +
and the ''umbral shift'', defined for a sequence $p_n(x)$ of binomial type by
 +
$$
 +
\theta p_n(x) = p_{n+1}(x) \ .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505036.png" /></td> </tr></table>
+
Two key results in the umbral calculus say that a linear operator on $P$ is an umbral operator if and only if its adjoint is an automorphism of $\mathcal{A}$, and an operator on $P$ is an umbral shift if and only if its adjoint is a derivation on $\mathcal{A}$. The first result leads to an explicit formula for the polynomials $p_n(x)$, and the second result leads to a recurrence relation for the $p_n(x)$, which gives well-known recurrences in the case of Hermite, Laguerre, Bernoulli, and other sequences.
  
and the umbral shift, defined for a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505037.png" /> of binomial type by
+
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 $q$-umbral calculus, where the polynomial coefficients are replaced by the Gaussian coefficients.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505038.png" /></td> </tr></table>
+
The Gaussian coefficients, or $q$-binomial coefficients $\left[{ \begin{array}{c} n \\ k \end{array} }\right]_q$, are defined by
 +
$$
 +
\left[{ \begin{array}{c} n \\ k \end{array} }\right]_q = \frac{ (q;q)_n }{ (q;q)_k (q;q)_{n-k} }
 +
$$
 +
where the so-called $q$-shifted factorials $(a;q)_m$ are defined by
 +
$$
 +
(a;q)_m = \begin{cases} 1 & \text{if}\, m=0 \\  (1-a)(1-qa) \cdots (1-q^{m-1}a) & \text{if}\, m>0 \ . \end{cases}
 +
$$
  
Two key results in the umbral calculus say that a linear operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505039.png" /> is an umbral operator if and only if its adjoint is an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505040.png" />, and an operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505041.png" /> is an umbral shift if and only if its adjoint is a derivation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505042.png" />. The first result leads to an explicit formula for the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505043.png" />, and the second result leads to a recurrence relation for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505044.png" />, which gives well-known recurrences in the case of Hermite, Laguerre, Bernoulli, and other sequences.
+
Here $q$ is seen either as a formal variable or as a complex variable of absolute value $< 1$. Using these $q$-binomial coefficients one has the $q$-binomial formula: If $x,y$ satisfy $y x = q x y$, then
 
+
$$
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505046.png" />-umbral calculus, where the polynomial coefficients are replaced by the Gaussian coefficients.
+
(x+y)^n = \sum_{k=0}^n \left[{ \begin{array}{c} n \\ k \end{array} }\right]_q x^{n-k} y^k = \sum_{k=0}^n \left[{ \begin{array}{c} n \\ k \end{array} }\right]_{q^{-1}} y^k x^{n-k} \ .
 
+
$$
The Gaussian coefficients, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505048.png" />-binomial coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505049.png" />, are defined by
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505050.png" /></td> </tr></table>
 
 
 
where the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505052.png" />-shifted factorials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505053.png" /> are defined by
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505054.png" /></td> </tr></table>
 
 
 
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505055.png" /> is seen either as a formal variable or as a complex variable of absolute value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505056.png" />. Using these <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505057.png" />-binomial coefficients one has the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505059.png" />-binomial formula: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505061.png" /> satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505062.png" />, then
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505063.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505064.png" /></td> </tr></table>
 
  
Currently a whole theory is emerging involving  "q-versions of various classical objects" : <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505065.png" />-special functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505066.png" />-gamma-function, quantum groups, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505067.png" />-integrals, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505068.png" />-orthogonal polynomials, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505069.png" />-hypergeometric series, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505070.png" />-Haar measure, etc., complete with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095050/u09505071.png" />-versions of the various interrelations between all these objects. Cf. also (the editorial comments to) [[Special functions|Special functions]]; [[Quantum groups|Quantum groups]] and [[#References|[a9]]]–[[#References|[a10]]].
+
Currently a whole theory is emerging involving  "$q$-versions of various classical objects" : $q$-special functions, $q$-gamma-function, quantum groups, $q$-integrals, $q$-orthogonal polynomials, $q$-hypergeometric series, $q$-Haar measure, etc., complete with $q$-versions of the various interrelations between all these objects. Cf. also (the editorial comments to) [[Special functions]]; [[Quantum groups]] and [[#References|[a9]]]–[[#References|[a10]]].
  
 
Finally, the umbral calculus has been generalized to study sequences of formal Laurent series, where the logarithm plays a key role.
 
Finally, the umbral calculus has been generalized to study sequences of formal Laurent series, where the logarithm plays a key role.
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<table>
 
<table>
 
<TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Loeb,  G.-C. Rota,  "Formal power series of logarithmic type"  ''Adv. Math.'' , '''75'''  (1989)  pp. 1–118</TD></TR>
 
<TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Loeb,  G.-C. Rota,  "Formal power series of logarithmic type"  ''Adv. Math.'' , '''75'''  (1989)  pp. 1–118</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Roman,  "The umbral calculus" , Acad. Press  (1984)</TD></TR>
+
<TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Roman,  "The umbral calculus" , Pure and Applied Mathematics '''111''', Acad. Press  (1984) {{ZBL|0536.33001}}</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Roman,  G.-C. Rota,  "The umbral calculus"  ''Adv. Math.'' , '''27'''  (1978)  pp. 95–188</TD></TR>
+
<TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Roman,  G.-C. Rota,  "The umbral calculus"  ''Adv. Math.'' , '''27'''  (1978)  pp. 95–188 {{ZBL|0375.05007}}</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Roman,  "More on the umbral calculus, with emphasis on the $q$-umbral calculus"  ''J. Math. Anal. Appl.'' , '''107'''  (1985)  pp. 222–254</TD></TR>
+
<TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Roman,  "More on the umbral calculus, with emphasis on the $q$-umbral calculus"  ''J. Math. Anal. Appl.'' , '''107'''  (1985)  pp. 222–254 {{ZBL|0654.05004}}</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Roman,  "The logarithmic binomial formula"  ''Amer. Math. Monthly'' (To appear)</TD></TR>
+
<TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Roman,  "The logarithmic binomial formula"  ''Amer. Math. Monthly'' '''99''', No.7 (1992) pp. 641-648 {{ZBL|0766.05006}}</TD></TR>
 
<TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Roman,  "The harmonic logarithms and the binomial formula"  ''J. Comb. Theory, Ser. A''  (To appear)</TD></TR>
 
<TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Roman,  "The harmonic logarithms and the binomial formula"  ''J. Comb. Theory, Ser. A''  (To appear)</TD></TR>
 
<TR><TD valign="top">[a7]</TD> <TD valign="top">  G.-C. Rota,  D. Kahaner,  A. Odlyzko,  "Finite operator calculus"  ''J. Math. Anal. Appl.'' , '''42'''  (1973)  pp. 684–760</TD></TR>
 
<TR><TD valign="top">[a7]</TD> <TD valign="top">  G.-C. Rota,  D. Kahaner,  A. Odlyzko,  "Finite operator calculus"  ''J. Math. Anal. Appl.'' , '''42'''  (1973)  pp. 684–760</TD></TR>
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</table>
 
</table>
  
{{TEX|part}}
+
{{TEX|done}}

Latest revision as of 20:34, 9 December 2015

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 $\mathbf{a}$ used to represent a sequence of real numbers $a_0,a_1,a_2,\ldots$. Thus, if the sequences $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ satisfy $$ c_n = \sum_{k=0}^n \binom{n}{k} a_k b_{n-k}\ , $$ this could be written in umbral notation as $$ \mathbf{c}^n = (\mathbf{a}+\mathbf{b})^n \ . $$ This notation is now obsolete, however.

The modern umbral calculus is designed to study polynomial sequences $p_n(x)$ of binomial type, that is, sequences for which $\deg p_n(x) = n$ and $$ p_n(x+y) = \sum_{k=0}^n \binom{n}{k} p_k(x) p_{n-k}(y) \ , $$ as well as polynomial sequences $s_n(x)$ of Sheffer type, that is, sequences for which $\deg s_n(x) = n$ and $$ s_n(x+y) = \sum_{k=0}^n \binom{n}{k} s_k(x) p_{n-k}(y) \ , $$ where $p_n(x)$ 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 $P$ is the algebra of polynomials in a single variable, then the dual space $P^*$ is well-known to be a vector space. In fact, $P^*$ is isomorphic to the vector space of formal power series $\mathcal{F}$ via the mapping $$ \sigma(L) = \sum_{k=0}^\infty \frac{ L(x^k) }{ k! } t^k \ . $$

One may therefore identify $P^*$ as the algebra $\sigma(P^*) = \mathcal{F}$, thus allowing for the multiplication of linear functionals. The algebra $\mathcal{A} = \sigma(P^*)$ is called the umbral algebra.

In particular, for linear functionals $L$ and $M$, the geometric series $M,ML,ML^2,\ldots$ makes sense, and so one may define a sequence $s_n(x)$ of polynomials by $\deg s_n(x) = n$ and the orthogonality conditions $$ ML^k(s_n(x)) = n! \delta_{kn} \ . $$ 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 $\mathcal{A}$. If $T : P \rightarrow P$ is a linear operator on $P$, its adjoint $T^* : \mathcal{A} \rightarrow \mathcal{A}$ is a linear operator on the umbral algebra $\mathcal{A}$.

The most important linear operators on $P$ are the umbral operator, defined for a sequence $p_n(x)$ of binomial type by $$ \lambda(x^n) = p_n(x) $$ and the umbral shift, defined for a sequence $p_n(x)$ of binomial type by $$ \theta p_n(x) = p_{n+1}(x) \ . $$

Two key results in the umbral calculus say that a linear operator on $P$ is an umbral operator if and only if its adjoint is an automorphism of $\mathcal{A}$, and an operator on $P$ is an umbral shift if and only if its adjoint is a derivation on $\mathcal{A}$. The first result leads to an explicit formula for the polynomials $p_n(x)$, and the second result leads to a recurrence relation for the $p_n(x)$, 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 $q$-umbral calculus, where the polynomial coefficients are replaced by the Gaussian coefficients.

The Gaussian coefficients, or $q$-binomial coefficients $\left[{ \begin{array}{c} n \\ k \end{array} }\right]_q$, are defined by $$ \left[{ \begin{array}{c} n \\ k \end{array} }\right]_q = \frac{ (q;q)_n }{ (q;q)_k (q;q)_{n-k} } $$ where the so-called $q$-shifted factorials $(a;q)_m$ are defined by $$ (a;q)_m = \begin{cases} 1 & \text{if}\, m=0 \\ (1-a)(1-qa) \cdots (1-q^{m-1}a) & \text{if}\, m>0 \ . \end{cases} $$

Here $q$ is seen either as a formal variable or as a complex variable of absolute value $< 1$. Using these $q$-binomial coefficients one has the $q$-binomial formula: If $x,y$ satisfy $y x = q x y$, then $$ (x+y)^n = \sum_{k=0}^n \left[{ \begin{array}{c} n \\ k \end{array} }\right]_q x^{n-k} y^k = \sum_{k=0}^n \left[{ \begin{array}{c} n \\ k \end{array} }\right]_{q^{-1}} y^k x^{n-k} \ . $$

Currently a whole theory is emerging involving "$q$-versions of various classical objects" : $q$-special functions, $q$-gamma-function, quantum groups, $q$-integrals, $q$-orthogonal polynomials, $q$-hypergeometric series, $q$-Haar measure, etc., complete with $q$-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.

References

[a1] D. Loeb, G.-C. Rota, "Formal power series of logarithmic type" Adv. Math. , 75 (1989) pp. 1–118
[a2] S. Roman, "The umbral calculus" , Pure and Applied Mathematics 111, Acad. Press (1984) Zbl 0536.33001
[a3] S. Roman, G.-C. Rota, "The umbral calculus" Adv. Math. , 27 (1978) pp. 95–188 Zbl 0375.05007
[a4] S. Roman, "More on the umbral calculus, with emphasis on the $q$-umbral calculus" J. Math. Anal. Appl. , 107 (1985) pp. 222–254 Zbl 0654.05004
[a5] S. Roman, "The logarithmic binomial formula" Amer. Math. Monthly 99, No.7 (1992) pp. 641-648 Zbl 0766.05006
[a6] S. Roman, "The harmonic logarithms and the binomial formula" J. Comb. Theory, Ser. A (To appear)
[a7] G.-C. Rota, D. Kahaner, A. Odlyzko, "Finite operator calculus" J. Math. Anal. Appl. , 42 (1973) pp. 684–760
[a8] K. Ueno, "Umbral calculus and special functions" Adv. Math. , 67 (1988) pp. 174–229
[a9] G. Gasper, M. Rahman, "Basic hypergeometric series" , Cambridge Univ. Press (1990)
[a10] T.H. Koornwinder, "Orthogonal polynomials in connection with quantum groups" P. Nevai (ed.) , Orthogonal polynomials: theory and practice , Kluwer (1990) pp. 257–292
How to Cite This Entry:
Umbral calculus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Umbral_calculus&oldid=36878
This article was adapted from an original article by S. Roman (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article