View source for Bell polynomial

The Bell polynomials (studied extensively by E.T. Bell [[#References|[a2]]]) arise naturally from differentiating a [[Composite function|composite function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b1102501.png" /> times, but in this context they predate Bell since they are implicit in the work of F. Faà di Bruno [[#References|[a4]]]. Accounts of Faà di Bruno's formula, however, often fail to mention any connection with Bell polynomials. The polynomials also occur in other places without being referred to by name; in [[#References|[a1]]] (Table 24.2), for example, the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b1102502.png" /> are coefficients of partial Bell polynomials, but are not identified as such.

Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b1102503.png" /> and let

<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/b/b110/b110250/b1102504.png" /></td> </tr></table>

then by repeated application of the chain rule:

<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/b/b110/b110250/b1102505.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/b/b110/b110250/b1102506.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/b/b110/b110250/b1102507.png" /></td> </tr></table>

In general,

<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/b/b110/b110250/b1102508.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b1102509.png" /> is a homogeneous polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025010.png" /> and weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025011.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025012.png" />, known as a (partial) Bell polynomial (see [[#References|[a3]]] for a table for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025013.png" />); it has integral coefficients. Because of the homogeneity, for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025014.png" /> all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025015.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025016.png" />) can be determined uniquely even if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025017.png" /> are omitted. Hence the (complete) Bell polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025018.png" /> is usually defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025019.png" /> 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/b/b110/b110250/b11025020.png" /></td> </tr></table>

In [[#References|[a7]]], however, the term Bell polynomial is used for

<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/b/b110/b110250/b11025021.png" /></td> </tr></table>

so the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025022.png" /> are included in the definition.

The following definitions are also made: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025024.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025025.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025026.png" />.

Although the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025027.png" /> were introduced as derivatives, the Bell polynomials themselves, considered purely as polynomials in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025028.png" /> are independent of the initial functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025030.png" />. Hence information can be deduced from special choices such as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025031.png" />, which gives

<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/b/b110/b110250/b11025032.png" /></td> </tr></table>

An alternative approach which gives the same polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025033.png" /> is adopted in [[#References|[a3]]], where they are defined as coefficients in the expansion of the two-variable [[Generating function|generating function]]

<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/b/b110/b110250/b11025034.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/b/b110/b110250/b11025035.png" /></td> </tr></table>

This approach obviates the earlier assumption that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025036.png" /> are derivatives.

The generating function for the complete polynomials is

<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/b/b110/b110250/b11025037.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/b/b110/b110250/b11025038.png" /></td> </tr></table>

Explicit formulas are known for Bell polynomials and they are examples of partition polynomials (multivariable polynomials which can be expressed as a sum of monomials, where the sum is over a set of partitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025039.png" />; cf. also [[Partition|Partition]]). The partial polynomial

<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/b/b110/b110250/b11025040.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/b/b110/b110250/b11025041.png" /></td> </tr></table>

where the sum is over all partitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025042.png" /> into exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025043.png" /> non-negative parts, i.e., over all solutions in non-negative integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025044.png" /> of the two equations

<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/b/b110/b110250/b11025045.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/b/b110/b110250/b11025046.png" /></td> </tr></table>

Since, for each fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025047.png" />, there can be no parts of size greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025048.png" />, the formula is often stated in the simpler looking, but equivalent, form (where necessarily <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025049.png" />):

<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/b/b110/b110250/b11025050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</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/b/b110/b110250/b11025051.png" /></td> </tr></table>

where the sum is over all solutions in non-negative integers of the equations

<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/b/b110/b110250/b11025052.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/b/b110/b110250/b11025053.png" /></td> </tr></table>

The complete polynomial

<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/b/b110/b110250/b11025054.png" /></td> </tr></table>

where the sum is over all partitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025055.png" /> into arbitrarily many non-negative parts, i.e., over all non-negative integer solutions of the single equation

<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/b/b110/b110250/b11025056.png" /></td> </tr></table>

There are many recurrence relations for Bell polynomials, as well as formulas connecting them with other special polynomials and numbers; the following is a small selection, and others may be found in [[#References|[a3]]], [[#References|[a7]]], [[#References|[a8]]].

<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/b/b110/b110250/b11025057.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/b/b110/b110250/b11025058.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/b/b110/b110250/b11025059.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/b/b110/b110250/b11025060.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/b/b110/b110250/b11025061.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025063.png" /> are Stirling numbers of the first and second kinds (cf. [[Combinatorial analysis|Combinatorial analysis]]), respectively;

<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/b/b110/b110250/b11025064.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025065.png" /> is the [[Lah number|Lah number]];

<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/b/b110/b110250/b11025066.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025067.png" /> are the [[Bell numbers|Bell numbers]].

Combining equations (a1) and (a2) gives Faà di Bruno's formula for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025068.png" />th derivative of a composite function:

<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/b/b110/b110250/b11025069.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/b/b110/b110250/b11025070.png" /></td> </tr></table>

summed over all solutions in non-negative integers of

<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/b/b110/b110250/b11025071.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/b/b110/b110250/b11025072.png" /></td> </tr></table>

(For a generalization to functions of several variables, see [[#References|[a5]]].)

The formula can be used, in particular, to express functions of power series as power series. If

<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/b/b110/b110250/b11025073.png" /></td> </tr></table>

and

<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/b/b110/b110250/b11025074.png" /></td> </tr></table>

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/b/b110/b110250/b11025075.png" /></td> </tr></table>

and Faà di Bruno's formula can be used to find <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025076.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025077.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/b/b110/b110250/b11025078.png" /></td> </tr></table>

Hence, applying the formula and evaluating the result at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025079.png" /> gives

<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/b/b110/b110250/b11025080.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</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/b/b110/b110250/b11025081.png" /></td> </tr></table>

Thus, provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025082.png" />,

<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/b/b110/b110250/b11025083.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/b/b110/b110250/b11025084.png" /></td> </tr></table>

A special case of (a3) is used [[#References|[a6]]] to express the cumulants (semi-invariants, cf. [[Semi-invariant(2)|Semi-invariant]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025085.png" /> of a [[Probability distribution|probability distribution]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025086.png" /> in terms of its moments (cf. [[Moment|Moment]])

<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/b/b110/b110250/b11025087.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/b/b110/b110250/b11025088.png" /></td> </tr></table>

The probability generating function of the distribution is

<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/b/b110/b110250/b11025089.png" /></td> </tr></table>

and it is easy to show that

<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/b/b110/b110250/b11025090.png" /></td> </tr></table>

The cumulants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025091.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025092.png" />) and their exponential generating function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025093.png" /> are defined in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025094.png" /> 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/b/b110/b110250/b11025095.png" /></td> </tr></table>

Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025096.png" />, it follows from (a3) that

<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/b/b110/b110250/b11025097.png" /></td> </tr></table>

Similarly, starting from

<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/b/b110/b110250/b11025098.png" /></td> </tr></table>

and applying Faà di Bruno's formula with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b11025099.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b110250100.png" /> (and noting that in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b110250101.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110250/b110250102.png" />), the inverse relation expressing moments in terms of cumulants reduces to

<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/b/b110/b110250/b110250103.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Abramowitz,   I. Stegun,   "Handbook of mathematical functions" , Dover, reprint  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.T. Bell,   "Exponential polynomials"  ''Ann. of Math.'' , '''35'''  (1934)  pp. 258–277</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Comtet,   "Advanced combinatorics" , Reidel  (1974)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F. Faà di Bruno,   "Note sur une nouvelle formule du calcul différentiel"  ''Quart. J. Math.'' , '''1'''  (1855)  pp. 359–360</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I.J. Good,   "The multivariate saddlepoint method and chi-squared for the multinomial distribution"  ''Ann. Math. Stat.'' , '''32'''  (1961)  pp. 535–548</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E. Lukács,   "Applications of Faà di Bruno's formula in mathematical statistics"  ''Amer. Math. Monthly'' , '''62'''  (1955)  pp. 340–348</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J. Riordan,   "Combinatorial identities" , Wiley  (1968)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  S. Roman,   "The umbral calculus" , Acad. Press  (1984)</TD></TR></table>