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\bf
\centerline
{
Fully AUTOMATED Computerized Redux of Feller's (v.1) Ch. III
(and Much More!)
}
\rm
\bigskip
\centerline{ {\it
Doron 
ZEILBERGER}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Rutgers University (New Brunswick),
Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway,
NJ 08854-8019, USA.
%\break
{\eighttt zeilberg  at math dot rutgers dot edu} ,
\hfill \break
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg} .
Nov. 10, 2006.
Accompanied by Maple package {\eighttt FELLER}
downloadable from Zeilberger's website.
Supported in part by the NSF.
}
}
\qquad\qquad
{\it In memory of my academic great-grandfather, Willy FELLER
and my biological great-grandfather, Adolf PINNER}

{\bf Two (Truly) GREAT Great-Grandparents: Adolf PINNER and Willy FELLER}

No wonder I am such a good writer!
It runs in the {\it biological} family. My (biological)
great-grandfather, Adolf Pinner (1842-1909), in addition to being
a brilliant chemist -he discovered the structure
of Nictoine and coined the name {\it Pyrimidine}-
was also a great expositor, and his chemistry
books[P1][P2] were the standard textbooks in that subject 
between 1872 and 
1910 (and beyond) in Germany and elsewhere. He also
wrote a charming popular book[P3], {\it The Laws of the Phenomena
of Nature}, that my second-cousin, Sister Dr. Katherine Wolff
(who is also his great-grandchild, and also a great writer!)
is currently translating from German to English, and 
her translation is available from 
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/family/gesetze.html}.

No wonder I am such a good writer!
It runs in the {\it academic} family. My (academic)
great-grandfather, Willy Feller (1906-1970), in addition to being
a brilliant probabilist-he discovered the Feller process
and contributed significantly to the relation between Markov
processes and differential equations-
was also a great expositor, and his probability
books[F1][F2] were the standard textbooks in that subject
between 1950 and
1980 (and beyond) in the USA and elsewhere, until students got
dumber and needed more watered-down texts.

{\bf Patience}

So if I am such a good writer, how come I don't write
textbooks?  Well, it takes more than that.
Both my great-grandfathers had in abundance something that
I {\it don't} have: {\bf patience}. 
Both Feller and Pinner had lots and lots of drafts, and kept editing and
polishing them. 

{\bf The Legendary Chapter III}

My favorite chapter in 
Feller's classic[F1] is Chapter III.
It is chuckfull of clever reflective combinatorics, 
and waxes eloquent about the amazing discrete arcsine law that
is so counter-intuitive: Most people are either winners most
of the time or losers most of the time. However, if they break
even when they die, then it is equally likely that they
were winners half of the time as that they were winners all the time
or no time, or quarter of the time, etc.

\vfill\eject
{\bf Generating Functions}

The very first approach[CF], abandoned later, was to use
{\it generating functions}. Feller (and, even many people today)
considered it as an {\it analytical} method. In the old days, one
first used combinatorial or probabilistic arguments to
derive {\it recurrences} for the {\it sequence}. Then they
took the (ordinary or exponential) generating function of
the sequence, and showed that it satisfied a 
(differential and/or algebraic and/or functional) equation, then
one {\it solved} the equation for the generating function, and
if in luck was able to deduce {\it closed-form} expressions for
the coefficients.

{\bf Weight-Enumerators}

But P\'olya, Tutte and Sch\"utzeneberger showed us
that the above outline is stupid. First, generating functions
are better viewed as neither generating nor functions.
They are rather {\it formal power series} that are
{\it weight-enumerators} of combinatorial sets, and one can operate
with them directly. Also, with computers, we no longer care how
``messy'' these generating functions turn out to be, and
computers can {\it guess}, and then {\it a posteriori} prove,
the closed-form expressions.

{\bf Coin Tosses}

Feller considered sequences of coin tosses ($H=1, T=-1$), let's
call them $w$ of
length $n$ and was interested in the following parameters.


$a_1(w):=$ the number of losing times, 
in symbols:
$$
a_1(w):=\vert \{ i \, \vert \,  \sum_{j=1}^{i} w_j<0 \, \, or  \,\,
\sum_{j=1}^{i} w_j=0 \, \, and  \,\,
\sum_{j=1}^{i-1} w_j<0\} \vert \quad ,
$$

$a_2(w):=$ the number of times it was breaking-even, 
in symbols:
$$
a_2(w):=\vert \{ i \, \vert \, \sum_{j=1}^{i} w_j=0 \} \vert \quad ,
$$

$a_3(w):=$ the last break-even time,
in symbols:
$$
a_3(w):= max \{ i \, \vert \, \sum_{j=1}^{i} w_j=0 ,
\sum_{j=1}^{r} w_j>0 \,\, for \,\, r>i\} \quad ,
$$

$a_4(w):=$ number of sign-changes,
in symbols:
$$
a_4(w):=\vert \{ i \, \vert \,
\sum_{j=1}^{i-1} w_j<0 \, \, and \,\,\sum_{j=1}^{i+1} w_j>0 \, \, 
or  \,\,
\sum_{j=1}^{i-1} w_j>0 \, \, and \,\,\sum_{j=1}^{i+1} w_j<0
\} \vert \quad .
$$


Feller[F1] (Ch. III)
only considered each of these parameters {\it one at a time},
but {\it we} can consider the {\it grand generating function}
$$
F(z,t_1,t_2,t_3,t_4):=
\sum_{n=0}^{\infty} \sum_{i_1=0}^{n}
\sum_{i_2=0}^{n}\sum_{i_3=0}^{n}\sum_{i_4=0}^{n} 
A_n(i_1,i_2,i_3,i_4)  t_1^{i_1} t_2^{i_2} t_3^{i_3} t_4^{i_4} z^n \quad ,
$$
where $A_n(i_1,i_2,i_3,i_4)$ is the number of sequences
$w$ of length $n$ with $a_1(w)=i_1, a_2(w)=i_2,a_3(w)=i_3,a_4(w)=i_4$.
Equivalently, (and much better!)
$$
F(z,t_1,t_2,t_3,t_4):=\sum_{w \in \{-1,1\}^{*}} weight(w) \quad,
$$
where the {\it weight} of an arbitrary word in the alphabet
$\{-1,1\}$ is
$$
weight(w):= {t_1}^{a_1(w)}{t_2}^{a_2(w)}
{t_3}^{a_3(w)}{t_4}^{a_4(w)} z^{length(w)} \quad.
$$

Also let $C(z,t_1,t_2,t_3,t_4)$ be the analogous weight-enumerator
for those sequences that add to $0$ (i.e. broke-even at the end).

{\bf Deriving the Five-Variable Generating Function
ONCE and FOR ALL}

The beginning is as in [Z]. Let
$\phi(z)$ be the generating function, according to length,
of words in $\{-1 , 1\}$ that broke-even at the end but was
never losing. Then of course
$$
\phi=1+z^2\phi^2 \quad ,
$$
since such a sequence is either empty (weight $1$) or
is a {\it catenation} of an {\it irreducible} sequence
(one that only breaks-even at the beginning and the end)
[weight $z \phi(z)z$] and a shorter sequence [weight $\phi(z)$].
Using the 3000-year-old formula for the quadratic equation, one
can get $\phi(z)$ (and hence $\psi(z):=z^2\phi(z)$) {\it explicitly}.
I am resisting the temptation to write it down, since
why bother? the computer can do it for itself.

Let $\alpha(z)$ be the generating function (weight-enumerator),
according to length, 
of the set of sequences that except at the beginning
always had a {\it positive} amount (partial sum).

Let $\beta(z)$ be the generating function (weight-enumerator),
according to length, 
of the set of sequences that except at the beginning
always had a {\it negative} amount (partial sum).

By symmetry $\alpha=\beta$ and by factoring such a winning
sequence according to each time where it had the next largest
amount for the {\it last} time we immediately get
$$
\alpha(z)=\beta(z)={{z} \over {1-z-\psi(z)}} \quad.
$$

Now given a sequence, let's call 

$\bullet$ a segment between two
consecutive break-even times, where it was  {\it not losing}
in-between, a $P$-segment.

$\bullet$  a segment between two
consecutive break-even times, where it was {\it not winning}
in-between, an $N$-segment.

$\bullet$ a segment that started with $0$ dollars, but
was {\it winning} for ever after, a $P'$-segment.

$\bullet$ a segment that started with $0$ dollars, but
was {\it losing} for ever after, an $N'$-segment.

If you factor a break-even
sequence (i.e. with sum $0$) according to its sign-changes,
and look at it as a sequence of $P$'s and $N$'s,
we have the following {\it regular expression}
$$
empty \vee P(NP)^{*} \vee P(NP)^{*}N \vee N(PN)^{*} \vee N(PN)^{*}P \quad,
$$
whose generating function (alias weight-enumerator) (in $P,N,t_4$) is:
$$
\overline{C}=1+ P{{1} \over {1-t_4Nt_4P}}
+P{{1} \over {1-t_4Nt_4P}}t_4N
+N{{1} \over {1-t_4Pt_4N}}
+N{{1} \over {1-t_4Pt_4N}}t_4P \quad.
$$

If you factor an {\it arbitrary}
sequence according to its sign-changes,
and look at it as a sequence of $P$'s and $N$'s,
possibly followed by $N'$ or $P'$
we have the following {\it regular expression}
$$
empty \vee P(NP)^{*} \vee P(NP)^{*}N \vee N(PN)^{*} \vee N(PN)^{*}P \quad
$$
$$
\vee
P' \vee P(NP)^{*}P' \vee P(NP)^{*}NP' \vee N(PN)^{*}P' \vee N(PN)^{*}PP' \quad
$$
$$
\vee 
N' \vee P(NP)^{*}N' \vee P(NP)^{*}NN' \vee N(PN)^{*}N' \vee N(PN)^{*}PN' 
\quad .
$$

whose generating function (in $P,N,P',N',t_4$) is:
$$
\overline{F}=1+ P{{1} \over {1-t_4Nt_4P}}
+P{{1} \over {1-t_4Nt_4P}}t_4N
+N{{1} \over {1-t_4Pt_4N}}
+N{{1} \over {1-t_4Pt_4N}}t_4P 
$$
$$
+P'+ P{{1} \over {1-t_4Nt_4P}}P'
+P{{1} \over {1-t_4Nt_4P}}t_4Nt_4P'
+N{{1} \over {1-t_4Pt_4N}}t_4P'
+N{{1} \over {1-t_4Pt_4N}}t_4PP' 
$$
$$
+N'+ P{{1} \over {1-t_4Nt_4P}}t_4N'
+P{{1} \over {1-t_4Nt_4P}}t_4NN'
+N{{1} \over {1-t_4Pt_4N}}N'
+N{{1} \over {1-t_4Pt_4N}}t_4Pt_4N' \quad.
$$
Now substituting in $\overline{F}$,
$$
P=t_2 \psi(t_3 z)/(1-t_2\psi(t_3z)) \quad , \quad 
N=t_2 \psi(t_1t_3 z)/(1-t_2\psi(t_1 t_3z)) \quad , \quad
P'=\alpha(z) \quad, \quad N'=\alpha(t_1z) \quad.
$$
you would get an {\it explicit} (but ``large'') expression for
$F(z,t_1,t_2,t_3,t_4)$.

Similarly, plugging-in $\overline{C}$ the $P$ and $N$ above,
would give $C(z,t_1,t_2,t_3,t_4)$, the analog for those
sequences that add to $0$.

This is all done internally by the Maple package {\tt FELLER}
that easily proves Theorem III.4.1
(that the number of sequences of $\{-1,1\}$ of length 
$2n$ where the longest prefix that sums to $0$ is
$2k$ equals ${{2k} \choose {k}}{{2n-2k} \choose {n-k}}$),
Theorem III.4.2 
(that the number of sequences of $\{-1,1\}$ of length 
$2n$ that was winning exactly $2k$ times equals
${{2k} \choose {k}}{{2n-2k} \choose {n-k}}$ [the discrete
arc-sine law]), and Theorem III.9
(that the number of sequences of $\{-1,1\}$ of length 
$2n$ and that sum to $0$, that was winning exactly $2k$ times 
equals 
${{2n} \choose {n}}/(n+1)$ 
[the Chung-Feller theorem]).

These are proved in the Maple package {\tt FELLER} by
typing {\tt ThIII41();},  {\tt ThIII42();}, and
{\tt ThIII9();}, respectively.

It is possible to milk $F(z,t_1,t_2,t_3,t_4)$ 
and $C(z,t_1,t_2,t_3,t_4)$ 
much more. You can {\it easily} find
more refined enumerations, averages, variances, 
higher moments, covariances etc. etc.
{\bf You} are welcome to explore {\tt FELLER} on your own!

{\bf References}

[CF] K.L. Chung and W. Feller, {\it On fluctuations
in coin tossing}, Proc. Nat. Acad. Sci. USA {\bf 35}
(1949), 263-270.

[F1] William Feller, ``{\it An Introduction to Probability
Theory and Its Application}'', volume 1, three editions.
John Wiley and sons. First edition: 1950. Second edition:
1957. Third edition: 1968.

[F2] William Feller, ``{\it An Introduction to Probability
Theory and Its Application}'', volume 2, two editions.
John Wiley and sons. 

[P1] Adolf Pinner, ``{\it Repetitorium der Organischen Chemie}'',
Berlin, Verlag von Robert Oppenheim, eleven editions.
First edition: 1872. Eleventh edition: 1901.

[P2] Adolf Pinner, ``{\it Repetitorium der Anorganischen Chemie}'',
Berlin, Verlag von Robert Oppenheim, [at least] seven editions.
First edition: 1873. Seventh edition: 1887.

[P3] Adolf Pinner, ``{\it Die Gesetze der Naturerschienungen}'',
Das Wissen der Gegenward, LXVI band. Leipzig, A. Freitag, 1888.
English translation [in progress] by Sister Katherine
Wolff available from 
\hfill\break
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/family/gesetze.html}.

[Z] D. Zeilberger, {\it A new proof that there are $2^n$ ways
of tossing a coin $n$ times}. Personal Journal of Ekhad
and Zeilberger,1995.
\hfill\break
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/pj.html}.



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