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\begin{document}

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\theoremstyle{plain}
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\begin{center}
\vskip 1cm{\LARGE\bf 
Integer Sequences Connected to the Laplace \\
\vskip .12in
Continued Fraction and Ramanujan's Identity
}
\vskip 1cm
\large
Alexander Kreinin\\
Risk Analytics \\
IBM\\
185 Spadina Avenue \\
Toronto, ON M5T 2C6 \\
Canada\\
\href{mailto:alex.kreinin@ca.ibm.com}{\tt alex.kreinin@ca.ibm.com } \\
\end{center}

\vskip .2 in
\begin{abstract}
We consider integer sequences connected to the famous Laplace continued
fraction for the function $R(t)=\int_t^\iy\vf(x) \dd x/\vf(t)$, where
$\vf(t) = e^{-t^2/2}/\sqrt{2\pi}$ is the standard normal density.  We
compute the generating functions for these sequences and study their
relation to the  Hermite and Bessel polynomials.  Using the master
equation for the generating functions, we find  a new proof of the
Ramanujan identity.
\end{abstract}


\section{Introduction}\label{intro}
We consider two infinite  matrices, $\fP=\Vert p_{k,m} \Vert_{k,m\ge 0}$ and
$\fQ=\Vert q_{k,m}\Vert_{k,m\ge 0}$, defined as follows.
If $m&gt;k$ or $k-m \equiv 1$ (mod $2$),  then
$$
    p_{k,m}=q_{k, m}=0,\quad  k,m =0, 1, \dots. 
$$   
If $k=m+2n$  and $n\ge 0$ then
\begin{equation}
    p_{k,m} = \frac{k!}{m!\,\, 2^n\,\, n!}, \label{eq_form_pkm}
\end{equation}
and 
\begin{equation}
    q_{k,m}=\frac{\Bigl(\frac{k+m}2\Bigr)!}{m!} 2^{-n} \sum_{j=0}^n {{k+1}\choose j}.
\label{eq_form_qkm}
\end{equation}

The matrices $\fP$ and $\fQ$ are connected to the Laplace continued fraction \cite{Laplace}
\begin{equation}
    \cL(t) \bydef \cfrac{1}{t+\cfrac{1}{t+\cfrac{2}{t+\cfrac{3}{t+\cfrac{4}{\ddots}}}}} \,, \quad t&gt;0,
\label{eq_Cont_frac}
\end{equation}
for the function $R(t)\bydef \bar\Phi(t)/{\vf(t)} $,
 where
$\vf(t)$ is the standard normal density and $\bar\Phi(t)=\int_t^\iy\vf(s) \dd s$
is the tail of the standard normal distribution. The function, $R(t)$, is often called  the Mills
ratio,  after John Mills who tabulated it \cite{Mills} in $1929$. The Mills ratio
appears in the probability theory,
\cite{Dumbgen, Feller70, McKean, Ruben} in the context of  asymptotic expansions for 
the normal probability integral,
in statistical analysis \cite{KendStuart, Mills} and in the area of numerical analysis
\cite{Baricz, Kouba, Pinelis, Shenton} where the functional inequalities 
and irrational approximations for $R(t)$ are discussed.
 
In this paper we study the matrices $\fP$ and $\fQ$. In Section~\ref{sec_elm}, we show that  
certain integer sequences in   \cite{Sloan} are encapsulated in these matrices. In particular,
the triangular array \seqnum{A180048}  is described by (\ref{eq_form_qkm}).
In Section~\ref{sec_elm}, we also describe connections between the matrix $\fP$ and the 
coefficients of the  Hermite and Bessel polynomials \cite{Abram, Andrews}. 

 In Section~\ref{sec_Lap_Polyn}, we introduce and study the generating polynomials.
Despite the fact that the recurrence relations for the polynomials $P_k$ and $Q_k$ have been known for
a very long time\footnote{Some of these relations  were derived by Jacobi in \cite{Jacobi}.}, 
their coefficients were not systematically studied until recently.
The first analysis of these coefficients, published in \cite{Kouba},
 appeared only in $2006$, to the best of our knowledge\footnote{Our definition of the coefficients of the
polynomials $Q_k(t)$ is different from that in \cite{Kouba}.}.

The statements in the first three sections are elementary. Their proofs are left to the reader.
 In Section~\ref{sec_GF},  we derive
 the master equation linking together the generating functions of the Laplace polynomials 
 and the Laplace continued fraction, $\cL(t)$.  In Section~\ref{sec_Ident}, the master equation
is used for a short derivation of the famous identity discovered by Ramanujan:
 \begin{equation*}
\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{\ddots}}} =
-1 + \frac{1}{\sqrt{ \frac{e\pi}{2} } - \sum_{n=0}^\iy\limits \frac{1}{(2n+1)!!} }.
\end{equation*}

 \section{Matrices $\fP$ and $\fQ$}\label{sec_elm}
We shall start with the following recurrence equations for the elements of $\fP$ and $\fQ$.
\begin{proposition}\label{prop_rr}
The elements of the matrices $\fP$ and $\fQ$ satisfy
\begin{eqnarray}
       m\cdot p_{k,m} &amp;=&amp;  k\cdot p_{k-1, m-1},  \,\,\,\,\qquad \qquad\qquad  k\ge 1, 1\le m\le k, \label{eq_rec_pp} \\
       p_{k+1, m} &amp;=&amp; p_{k, m-1} + (m+1)\cdot p_{k, m+1}, \quad k\ge 0, 1\le m\le k,  \label{eq_rec_pkm}  \\
q_{k, m} &amp;=&amp; p_{k, m} + (m+1)\cdot q_{k-1, m+1}, \quad k\ge 1, 1\le m\le k,  \label{eq_rec_qkm} \\
q_{k, m} &amp;=&amp; q_{k-1, m-1} + k\cdot q_{k-2, m}, \,\,\,\,\, \qquad k\ge 2, 1\le m\le k.  \label{eq_rec_qkmL}
\end{eqnarray}
\end{proposition}

\begin{proof}
The proof is based on the following lemma.
\begin{lemma}\label{lem_bnc}
Write
$$ S_i^{(j)}(k) = \sum_{l=i}^j {k\choose l}, \quad k&gt;j. $$
Then
\begin{equation} S_0^{(n-1)}(k) - 2 S_0^{(n-2)}(k-1) = {{k-1}\choose {n-1}}. 
\label{eq_lem_bnc}
\end{equation}
\end{lemma}
The derivation of (\ref{eq_rec_pp}) is straightforward.
The derivation of  (\ref{eq_rec_pkm}),  (\ref{eq_rec_qkm}) and
  (\ref{eq_rec_qkmL}) is based on Lemma~\ref{lem_bnc},
which in the case of  (\ref{eq_rec_qkmL}) is used twice. 
\end{proof}
  
The elements in the first nine rows and columns of the matrix $\fP$ are given in Table~\ref{tab_pkm}.

\begin{table}[H]
\begin{center}
\begin{tabular}{||c||c|c|c|c|c|c|c|c|c||}
\hline
  $k$  &amp; \multicolumn{9}{|c|}{$m$}   \\ \cline{2-10}
       &amp; $0$   &amp; $1$  &amp; $2$ &amp; $3$ &amp; $4$ &amp; $5$ &amp; $6$ &amp; $7$ &amp; $8$  \\ \hline \hline
 $0$  &amp; $1$  &amp; $0$ &amp;  $0$  &amp;  $\dots$   &amp;    &amp;   &amp;   &amp;      &amp;  $0$     \\ \hline
 $1$  &amp; $0$  &amp; $1$ &amp;  $0$  &amp;  $\dots$   &amp;    &amp;   &amp;   &amp;      &amp;  $0$     \\ \hline
 $2$  &amp; $1$  &amp; $0$ &amp; $1$ &amp;  $0$ &amp; $\dots$ &amp;     &amp;     &amp;     &amp; $0$      \\ \hline
 $3$  &amp; $0$  &amp; $3$ &amp; $0$ &amp; $1$ &amp; $0$ &amp; $\dots$ &amp;     &amp;      &amp;  $0$       \\ \hline
 $4$  &amp; $3$  &amp; $0$  &amp; $6$ &amp; $0$  &amp; $1$ &amp; $0$ &amp; $\dots$  &amp;  &amp; $0$             \\ \hline
 $5$  &amp; $0$  &amp; $15$ &amp; $0$ &amp; $10$ &amp; $0$ &amp; $1$ &amp; $0$ &amp;  $\dots$   &amp; $0$         \\ \hline
 $6$  &amp; $15$ &amp; $0$  &amp; $45$ &amp; $0$ &amp; $15$ &amp; $0$ &amp; $1$ &amp;  $0$   &amp;  $0$        \\ \hline
 $7$  &amp; $0$ &amp; $105$  &amp; $0$ &amp; $105$ &amp; $0$ &amp; $21$ &amp; $0$&amp; $1$ &amp;    $0$       \\ \hline
 $8$  &amp; $105$ &amp; $0$ &amp; $420$ &amp; $0$ &amp; $210$ &amp; $0$ &amp; $28$ &amp; $0$ &amp; $1$  \\ \hline\hline
\end{tabular}
\end{center}
\begin{center}
\begin{minipage}{6cm }
\caption{Matrix $\fP=\Vert p_{k,m}\Vert$. }\label{tab_pkm} 
\end{minipage}\end{center}
\end{table}

The matrix $\fP$ encapsulates many remarkable integer sequences. The rows of the matrix describe the coefficients of the Laplace
polynomials \cite{Krein}, $P_k(t)=\sum_{m=0}^k\limits p_{k,m} t^m$. These polynomials  will be discussed in the next section. 
The columns of $\fP$ are integer sequences that can be found in \cite{Sloan}. In particular, 
the first column is $p_{2n,0}=(2n-1)!!$, $(n=1, 2, \dots)$, which is \seqnum{A001147}. In the case $m=1$, the column 
$p_{2n-1, 1}=p_{2n,0}$. If $m=2$ then $p_{2n+2, 2}= (n+1) \cdot (2n+1)!!$, which is \seqnum{A001879}.
In the case $m=3$ we have the sequence \seqnum{A000457}.

The diagonals of $\fP$ also represent some remarkable integer sequences.
The diagonal $p_{k,2N-k}$ represents the coefficients of the $N$th Bessel polynomial. 
Indeed, the Bessel polynomials are
$$  y_N(t) = \sum_{j=0}^N \frac{(N+j)!}{(N-j)! \cdot j!} \Bigl( \frac{t}{2}\Bigr)^j, \quad N=0, 1, \dots. $$
Their coefficients, $B_{N,j}\bydef (N+j)!/((N-j)! j! 2^j)$, are equal to $p_{N+j, N-j}$, for $j=0, 1, \dots, N$.
Thus the Bessel polynomials can be written as 
 $y_N(t) = \sum_{j=0}^N\limits  p_{N+j, N-j}t^j$. 

The diagonals $k-m=2n$ also represent some well-known sequences. If $n=0$ then we have the sequence \seqnum{A000012}.
If $n=1$, $p_{m+2, m}=(m+1)(m+2)/2$ is the sequence of triangular numbers \seqnum{A000217}.
If $n=2$, $p_{m+4, m}$ is the sequence of tritriangular numbers \seqnum{A050534}.

Analytical properties of the matrix $\fQ$  are equally  interesting.
The elements in the first eight rows  of the matrix $\fQ$ are given in Table~\ref{tab_qkm}.
For small $m$, Formula~(\ref{eq_form_qkm}) can be simplified. 

\begin{table}[H]
\begin{center}
\begin{tabular}{||c||c|c|c|c|c|c|c|c|c||}
\hline
  $k$  &amp; \multicolumn{9}{|c|}{$m$}   \\ \cline{2-10}
       &amp; $0$   &amp; $1$  &amp; $2$ &amp; $3$ &amp; $4$ &amp; $5$ &amp; $6$ &amp; $7$ &amp; $8$  \\ \hline \hline
 $0$  &amp; $1$  &amp; $0$ &amp;  $0$  &amp;  $\dots$   &amp;    &amp;   &amp;   &amp;      &amp;  $0$     \\ \hline
 $1$  &amp; $0$  &amp; $1$ &amp;  $0$  &amp;  $\dots$   &amp;    &amp;   &amp;   &amp;      &amp;  $0$     \\ \hline
 $2$  &amp; $2$  &amp; $0$ &amp; $1$ &amp;  $0$ &amp; $\dots$ &amp;     &amp;     &amp;     &amp; $0$      \\ \hline
 $3$  &amp; $0$  &amp; $5$ &amp; $0$ &amp; $1$ &amp; $0$ &amp; $\dots$ &amp;     &amp;      &amp;  $0$       \\ \hline
 $4$  &amp; $8$  &amp; $0$  &amp; $9$ &amp; $0$  &amp; $1$ &amp; $0$ &amp; $\dots$  &amp;  &amp; $0$             \\ \hline
 $5$  &amp; $0$  &amp; $33$ &amp; $0$ &amp; $14$ &amp; $0$ &amp; $1$ &amp; $0$ &amp;  $\dots$   &amp; $0$         \\ \hline
 $6$  &amp; $48$ &amp; $0$  &amp; $87$ &amp; $0$ &amp; $20$ &amp; $0$ &amp; $1$ &amp;  $0$   &amp;  $0$        \\ \hline
 $7$  &amp; $0$ &amp; $279$  &amp; $0$ &amp; $185$ &amp; $0$ &amp; $27$ &amp; $0$&amp; $1$ &amp;    $0$       \\ \hline
\end{tabular}
\end{center}
\begin{center}
\begin{minipage}{8cm }
\caption{Elements  of the matrix $\fQ=\Vert q_{k, m}\Vert$.}\label{tab_qkm}
\end{minipage}\end{center}
\end{table}

\begin{proposition}\label{cor_q_k0} For  $n=0, 1, 2, \dots$, we have
\begin{eqnarray}
q_{2n, 0} &amp;=&amp; (2n)!!, \label{eq_q_k0} \\
q_{2n+1, 1} &amp;=&amp; (2n+2)!! - (2n+1)!!,  \label{eq_q_2n1} \\
q_{2n+2, 2} &amp;=&amp; \frac12 (2n+4)!! - (2n+3)!!, \label{eq_q_2n2} \\
q_{2n+3, 3} &amp;=&amp; \frac1{3!} (2n+6)!! - \frac{1}{2!} (2n+5)!! + \frac{1}{3!}(2n+3)!!, \label{eq_q_2n3} \\
q_{2n+4, 4} &amp;=&amp; \frac{1}{4!} (2n+8)!! - \frac{1}{3!} (2n+7)!! + \frac{1}{3!}(2n+5)!!. \label{eq_q_2n4}
\end{eqnarray}
\end{proposition}
According to Proposition~\ref{cor_q_k0}, the first column of the matrix $\fQ$, described by (\ref{eq_q_k0}),
 corresponds to the sequence \seqnum{A000165}. 
The second column, satisfying Equation~(\ref{eq_q_2n1}), describes the sequence \seqnum{A129890}. 
The third column, satisfying Equation~(\ref{eq_q_2n2}), represents the integer sequence \seqnum{A035101}. 
This sequence is related to the Catalan numbers (see \cite{Sloan}) but our formula (\ref{eq_q_2n2}) looks simpler.
The sequence $q_{2n+3, 3}$ represents the integer sequence \seqnum{A263384}.
The sequence $q_{2n+4, 4}$,  described by (\ref{eq_q_2n4}), is less known. 

The elements of the matrix $\fQ=\Vert q_{k, m}\Vert$ can be represented as a linear combination of
 the diagonal elements of the  matrix $\fP$.
Let  $k\equiv m$  (mod $2$), and $ n=(k-m)/2$. Then \cite{Krein}
$$
    m!\cdot q_{k, m}=\sum_{j=0}^n  (m+j)!\cdot p_{k-j, m+j}.
$$
   
\section{Polynomials $P_k(t)$ and $Q_k(t)$}\label{sec_Lap_Polyn}
\subsection{Laplace and Jacobi polynomials}
Consider the polynomials
$ P_k(t)\bydef \sum_{m=0}^k\limits p_{k,m} t^m, \quad
\text{and} \quad
 Q_k(t)\bydef \sum_{m=0}^k\limits q_{k,m} t^m$,
where $p_{k,m}$ and $q_{k,m}$ are defined by (\ref{eq_form_pkm}) and (\ref{eq_form_qkm}), respectively.
In this section, we shall connect these polynomials with the convergents of the 
Laplace continued fraction, $\cL(t)$.
For this reason, we  call 
$P_k(t)$ and $Q_k(t)$ the {\it Laplace polynomials} in
what follows.
\begin{proposition}[\cite{Jacobi, Kouba, Krein, Pinelis}] \label{prop_pk}
For $k\ge 1$, the Laplace  polynomials satisfy the following recurrent equations
\begin{eqnarray}
P_{k+1}(t) &amp;=&amp; t P_k(t) + P_k^\prime(t), \label{eq_P_k}\\
Q_k(t) &amp;=&amp; P_k(t) + Q_{k-1}^\prime(t),   \label{eq_Q_k} \\
P_{k+1}(t) &amp;=&amp; t P_k(t) + k P_{k-1}(t), \label{eq_P_kS}\\
Q_{k+1}(t) &amp;=&amp; t Q_k(t) + (k+1) Q_{k-1}(t), \label{eq_Q_kS}
\end{eqnarray}
where $P_0(t)=Q_0(t)=1$.
\end{proposition}
Proposition~\ref{prop_pk} allows one to find the polynomials 
 $P_k(t)$ and $Q_k(t)$ for any integer $k\ge 1$.     
The first eight polynomials are given in Table~\ref{table_poly}.

\begin{table}[H]
\begin{center}
\begin{tabular}{||c||c||c||}
\hline
  $k$  &amp; $P_k(t)$   &amp; $Q_{k-1}(t)$ \\ \hline
   $1$  &amp; $t$  &amp; $1$ \\ \hline
   $2$ &amp; $t^2+1$ &amp; $t$ \\ \hline
   $3$ &amp; $t^3+3t$  &amp; $t^2+2$ \\ \hline
   $4$ &amp; $t^4+6t^2+3$  &amp; $t^3+5t$ \\ \hline
   $5$ &amp; $t^5+10t^3 +15t$  &amp; $t^4+9 t^2 +8$ \\ \hline
   $6$ &amp; $t^6+15t^4 + 45 t^2 + 15$ &amp; $t^5 + 14 t^3 + 33 t $  \\ \hline
   $7$ &amp; $t^7+21t^5 + 105 t^3 + 105t$  &amp; $t^6 + 20 t^4 + 87 t^2 +48  $ \\ \hline
   $8$ &amp; $t^8 + 28t^6 + 210 t^4 + 420 t^2 + 105$ &amp; $t^7 + 27t^5+185t^3 +279t  $  \\ 
\hline\hline
\end{tabular}
\end{center}
\begin{center}
\begin{minipage}{9cm }
\caption{Laplace polynomials $P_k(t)$ and $Q_{k-1}(t)$.}\label{table_poly}
\end{minipage}\end{center}
\end{table}
For $t&gt;0$, we consider the rational functions
$ \cL_k(t) = Q_{k-1}(t)/P_k(t),  (k=1, 2, \dots)$. 
The function $\cL_k(t)$ is the $k$th convergent of the continued 
fraction\footnote{See \cite{Kouba} and \cite{Krein} regarding this statement}, $\cL(t)$:
$$ \lim_{k\to\iy} \cL_k(t)=\cL(t)=R(t), \quad t&gt;0. $$
 
The function $R(t)$ satisfies the differential equation
\begin{equation}
    \frac{\dd  R(t)}{\dd t}= t\cdot R(t) -1. 
\label{eq_DE_hf}
\end{equation}
Let us now consider the derivatives of the function $R(t)$. From (\ref{eq_DE_hf}) we derive
\begin{equation}
\frac{{\mathrm{d}}^k R(t)}{ {\mathrm{d}}t^k}= t\cdot \frac{{\mathrm{d}}^{k-1} R(t)}
{{\mathrm{d}} t^{k-1}} +
(k-1)\cdot \frac{d^{k-2} R(t)}{{\mathrm{d}} t^{k-2}}.
\quad k=2, 3, \dots.
\label{eq_mrec_drvR} 
\end{equation}
It was proved in \cite{Jacobi, Kouba, Krein} that 
\footnote{Equation~(\ref{eq_fEq_LaplPoly}) follows from (\ref{eq_DE_hf}), (\ref{eq_mrec_drvR}) and Proposition~\ref {prop_pk}.} 
 
\begin{equation}
\frac{{\mathrm{d}}^k R(t)}{{\mathrm{d}} t^k}= R(t)\cdot P_k(t) 
   -  Q_{k-1}(t), \qquad k=1, 2, \dots.
\label{eq_fEq_LaplPoly} \end{equation}
\vskip 0.2cm

\subsection{Laplace and Hermite polynomials}\label{sec_LH}
 
The polynomials $P_k(t)$ are closely connected to the Hermite polynomials \cite{Abram, Andrews}.
Denote  the  differentiation operator by $\fD$: $\fD g(t)= \frac{\dd g(t)}{\dd t}$. Then, as usual,  
 $\fD^n g(t)= \frac{\dd^n g(t)}{\dd t^n}$, $(n=1, 2, \dots)$.
Recall that the Hermite  polynomials can be defined as 
$$ 
   H_k(t) \bydef (-1)^k e^{t^2/2} \, \fD^k  e^{-t^2/2}.
$$
\begin{lemma}\label{lem_Lp_df}
The Laplace polynomials, $P_k(t)$, satisfy
\begin{equation}\label{eqn_Lp_dif_form}  
   P_k(t) = e^{-t^2/2} \,\fD^k e^{t^2/2}, \quad k=0, 1, 2, \dots.
\end{equation}
\end{lemma}

\begin{proof}
We shall prove this lemma by induction.
For $k=1$
$$ e^{-t^2/2} \,\fD e^{t^2/2} = t=P_1(t). $$
Suppose that $e^{-t^2/2} \,\fD^n e^{t^2/2} = P_n(t)$.
Let us verify that  
$$ P_{n+1}(t) = e^{-t^2/2} \,\fD^{n+1} e^{t^2/2}. $$ 
Indeed,
\begin{eqnarray*}
    \fD^{n+1}  e^{t^2/2} &amp;=&amp; \fD\Bigl( \fD^n e^{t^2/2}\Bigr) \\
     &amp;=&amp; \fD\Bigl(  e^{t^2/2}\cdot P_n(t)\Bigr) \\
      &amp;=&amp; e^{t^2/2}\,\Bigl( t P_n(t) + P_n^\prime(t)\Bigr).
\end{eqnarray*}
Therefore
$$ e^{-t^2/2} \,\fD^{n+1} e^{t^2/2}=t P_n(t) + P_n^\prime(t). $$
Since Equation~(\ref{eq_P_k}) and the initial condition, $P_1(t)=t$, uniquely determine the sequence of 
polynomials generated by the recurrence (\ref{eq_P_k}), we obtain 
$$ e^{-t^2/2} \,\fD^{n+1} e^{t^2/2}=P_{n+1}(t),$$ as was to be
proved.  
\end{proof}

Lemma~\ref{lem_Lp_df} implies (see \cite{Kouba, Krein})
\begin{equation} 
    P_k(t) = (-\mathit i)^k H_k( {\mathit i} t).   \label{eq_P_H_k}
\end{equation}
This relation  allows us to reformulate the
classical  results obtained for the Hermite polynomials in terms of
the Laplace polynomials. In particular, one can easily derive from (\ref{eq_P_H_k}) and 
the generating function of the Hermite polynomials,
$$ \cH(t, s)\bydef \sum_{k=0}^\iy H_k(t) \frac{s^k}{k!}=e^{st-s^2/2} \, , $$
the generating function
$\cP(s, t) =\sum_{k=0}^\iy P_k(t) \frac{s^k}{k!}$.
\begin{lemma}[\cite{Kouba, Krein}]
The generating function, $\cP(s, t)$, is
\begin{equation}
    \cP(s, t) =\exp\Bigl( st + \frac{s^2}{2}\Bigr).   \label{eq_gen_funct}
\end{equation}
\end{lemma}
 
\section{Master equation}\label{sec_GF}
Let us now compute  the generating function of the Laplace polynomials $Q_k(t)$.
Write
$$ \cQ(s, t)\bydef \sum_{k=0}^\iy \sum_{m=0}^\iy q_{k,m} t^m \frac{s^{k+1}}{(k+1)!}. $$
\begin{lemma}\label{lem_gf_q}
The generating function $\cQ(s, t)$ is
\begin{equation}
\cQ(s, t) = \sqrt{2\pi}\, e^{(s+t)^2/2}\cdot \Bigl(\bar\Phi(t)- \bar\Phi(s+t) \Bigr).
\label{eq_gfQ}
\end{equation}
\end{lemma}

\begin{proof}
We have, 
$$ \cQ(s, t) = \sum_{k=0}^\iy Q_{k}(t)\frac{s^{k+1}}{(k+1)!} \quad \text{and}\quad
    \cP(s, t) = \sum_{k=0}^\iy P_{k}(t) \frac{s^k}{k!}\, .
$$
The Taylor series expansion for the function $R(t)$
 can be written as
$$ R(s+t) = \sum_{k=0}^\iy  \frac{\dd^k R(t)}{\dd t^k}\frac{s^k}{k!} .$$
Recall that the derivatives of the function $R(t)$ satisfy Equation~(\ref{eq_fEq_LaplPoly}):
$$\frac{ \dd^k R(t)}{\dd t^k} = P_k(t) R(t) -  Q_{k-1}(t). $$
 We have
$$ R(s+t) = \sqrt{2\pi } e^{(s+t)^2/2} \bar\Phi(s+t)\, . $$
Therefore
\begin{eqnarray*} 
    \sqrt{2\pi }\, e^{(s+t)^2/2}\cdot \bar\Phi(s+t) &amp;=&amp; \sum_{k=0}^\iy  
\frac{\dd^k R(t)}{\dd t^k}\frac{s^k}{k!}  \\
       &amp;=&amp; R(t) \sum_{k=0}^\iy P_k(t) \frac{s^k}{k!} - 
    \sum_{k=0}^\iy Q_{k-1}(t)  \frac{s^k}{k!} \\
  &amp;=&amp; R(t) \cdot e^{st+s^2/2} - \cQ(s, t) \\
 &amp;=&amp; \sqrt{2\pi}\, e^{t^2/2}\cdot \bar\Phi(t) \cdot e^{st+s^2/2} - \cQ(s, t).
\end{eqnarray*}   
Finally, we obtain 
$$  \cQ(s, t) =\sqrt{2\pi}\, e^{(s+t)^2/2}\cdot\Bigl(\bar\Phi(t) -  \bar\Phi(s+t)\Bigr). $$
Equation (\ref{eq_gfQ}) is thus proved.     
\end{proof}

\begin{theorem} \label{thm_meq}
The  generating functions, $\cP(s, t)$ and $\cQ(s,t)$, satisfy 
\begin{equation} 
    R(s+t) + \cQ(s, t) = \cP(s,t) R(t).   \label{eq_AltQ}
\end{equation}
\end{theorem}

\begin{proof}

We have $\sqrt{2\pi}\, \exp(t^2/2)\cdot \bar\Phi(t) = R(t)$.
From Lemma~\ref{lem_gf_q} we derive
\begin{eqnarray*}
\cQ(s, t) 
 &amp;=&amp; \sqrt{2\pi}\, e^{(s+t)^2/2}\cdot \Bigl(\bar\Phi(t) - \bar\Phi(s+t) \Bigr) \\
 &amp;=&amp;  e^{st+s^2/2} \cdot R(t) - R(t+s). 
\end{eqnarray*}
The latter relation is equivalent to (\ref{eq_AltQ}).
\end{proof}


\section{Identities}\label{sec_Ident}
In \cite{Krein} we derived a new, to the best of our knowledge, identity
$$
    \sum_{n=0}^\iy \sum_{m=0}^\iy \sum_{j=0}^{n}
    \frac{(n+m)!}{m! j! (m+2n+1-j)!}\,\, 2^{-n}  
    =\sqrt{2\pi} e^2 \Bigl(\Phi(2)-\Phi(1)\Bigr), 
$$
having an unusual combination of constants in the right hand side. 
In fact, the whole derivation is based on the  substitution $s=t=1$ into (\ref{eq_AltQ})
in Theorem~\ref{thm_meq}. Choosing  integer values for $s$ and $t$ we obtain a series of elegant
combinatorial identities similar to those discussed in \cite{Krein}.
Another interesting, unexpected  connection to the Laplace continued fraction 
is the Ramanujan identity  \cite[Entry 43, p.~166]{BRaman}:
 \begin{equation}
\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\cfrac{4}{1+\cfrac{5}{\ddots}}}}} =
 \frac{1}{\sqrt{ \frac{e\pi}{2} } - \sum_{n=0}^\iy\limits \frac{1}{(2n+1)!!} } -1.
\label{eq_ramanu}
\end{equation}
This identity can be derived from Equation~(\ref{eq_AltQ}). We will need the following  
generalization of  (\ref{eq_ramanu}).
\begin{proposition}\label{prop_Ram}
For $s&gt;0$, consider the continued fraction
$$ \cS(s)=   \cfrac{1}{s+\cfrac{2}{s+\cfrac{3}{s+\cfrac{4}{s+\cfrac{5}{\ddots}}}}} \,. $$
Then 
\begin{equation}
\cS(s) 
= \frac{1}{ e^{{s^2}/{2}}\sqrt{ \frac{\pi}{2} } - \sum_{n=0}^\iy\limits\frac{s^{2n+1}}{(2n+1)!!} } - s.
\label{eq_cf_fin} 
\end{equation}
\end{proposition}

\begin{proof}
From (\ref{eq_AltQ}) we derive
\begin{equation} 
    R(s) + \cQ(s, 0) = e^{s^2/2} \cdot R(0), \label{eq_RQ1}
\end{equation}
and
\begin{eqnarray} 
\cQ(s, 0) &amp;=&amp; \sum_{n=0}^\iy q_{2n, 0} \frac{s^{2n+1}}{(2n+1)!} \nonumber \\
 &amp;=&amp; \sum_{n=0}^\iy (2n)!! \frac{s^{2n+1}}{(2n+1)!} \nonumber \\
&amp;=&amp; \sum_{n=0}^\iy  \frac{s^{2n+1}}{(2n+1)!!}. \label{eq_cQ_0}  
\end{eqnarray}
Inspecting the continued fractions (\ref{eq_Cont_frac}) and $\cS(s)$ we find  
\begin{equation}
\cS(s) = -s + \frac{1}{\cL(s)}, \label{eq_Ut}
\end{equation} 
and, therefore, for $s&gt;0$
$$
\cS(s) = -s + \frac{1}{R(s)}.
$$
Taking into account that $R(0)=\sqrt{\pi/2}$ we obtain from  the latter equation,
 (\ref{eq_RQ1}) and  (\ref{eq_cQ_0})
\begin{equation}
\frac{1}{s+ \cS(s)} = e^{{s^2}/{2}}\sqrt{ \frac{\pi}{2} } - \sum_{n=0}^\iy \frac{s^{2n+1}}{(2n+1)!!}. 
\label{eq_Cs}
\end{equation}
Equation~(\ref{eq_Cs}) is equivalent to (\ref{eq_cf_fin}). 
\end{proof}

Substituting $s=1$ into  (\ref{eq_cf_fin}) we obtain the Ramanujan identity (\ref{eq_ramanu}).

\begin{remark}
The series $\sum_{n=0}^\iy\limits  \frac{s^{2n+1}}{(2n+1)!!}$ converges to the function
$$ e^{s^2/2}\int_0^s e^{-u^2/2} \dd u =(\Phi(s)-0.5)/\vf(s), $$
where $\vf(s)=\exp(-s^2/2)/\sqrt{2\pi}$.
The coefficients of the convergents of the continued fraction $\cS(s)$ are described by the triangular array
\seqnum{A180048}. 
\end{remark}
 
\section{Acknowledgments}I am very  grateful to Ian Iscoe, Sebastian Jaimungal,  
Alexey Kuznetsov, Tom Salisbury, Eugene Seneta, 
Isaac Sonin, Hans Tuenter, Alek Vainshtein, Vladimir Vinogradov,
and Vladimir Zhuravlev for interesting comments 
and stimulating discussions.  I would also like to thank  the referee for the suggestions and critical comments.
This paper would never appear without the OEIS.

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G. Andrews, R. Askey, and R. Roy, \emph{Special Functions}, 
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A. Baricz,  Mills' ratio: Monotonicity patterns and functional
  inequalities, \emph{J. Math. Anal. Appl. } \textbf{340} (2008),
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B. C. Berndt, \emph{Ramanujan's Notebooks}, Vol.~2,
Springer-Verlag, 1989.

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L. D\H{u}embgen,  Bounding standard Gaussian tail probability. 
University of Bern,
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W. Feller, \emph{An Introduction to Probability Theory and its Applications},
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C.~G.~J. Jacobi, De fractione continua, in quam integrale, $\int_x^\iy
  e^{-xx}\dd x$ evolvere licet, \emph{J. Reine Angew. Math.}
  \textbf{12} (1834), 346--347.

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M. Kendall and A. Stuart,  \emph{The Advanced Theory of Statistics}, \textbf{1},
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\bibitem{Kouba}
O. Kouba, Inequalities related to the error function, preprint,
\url{http://arXiv:math/0607694}, 2006.

\bibitem{Krein}
A. Kreinin,  Combinatorial properties of Mills' ratio, preprint,
 \url{http://arXiv.org/abs/1405.5852}, 2014.

\bibitem{Laplace}
P.~S. Laplace, \emph{Trait\'e de M\'ecanique Celeste}, Vol.~4, Courcier, 1805.

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H. McKean, \emph{Stochastic Integrals}, Academic Press, 1969.

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J. Mills, Table of the ratio area to bounding ordinate for any portion
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I. Pinelis,  Monotonicity properties of the relative error of a Pad\'e
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\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2000 {\it Mathematics Subject Classification}:
Primary 11Y05; Secondary 11Y55.

\noindent \emph{Keywords:} continued fraction,
integer sequence, Ramanujan identity.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A000012},
\seqnum{A000165},
\seqnum{A000217},
\seqnum{A000457},
\seqnum{A001147},
\seqnum{A001879},
\seqnum{A035101},
\seqnum{A050534},
\seqnum{A129890},
\seqnum{A180048}, and
\seqnum{A263384}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received November 11 2015; revised versions received
April 5 2016; June 24 2016; June 27 2016.
Published in {\it Journal of Integer Sequences}, June 27 2016.

\bigskip
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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.math.uwaterloo.ca/JIS/}.
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