# §2.10 Sums and Sequences

## §2.10(i) Euler–Maclaurin Formula

As in §24.2, let $B_{n}$ and $B_{n}\left(x\right)$ denote the $n$th Bernoulli number and polynomial, respectively, and $\widetilde{B}_{n}\left(x\right)$ the $n$th Bernoulli periodic function $B_{n}\left(x-\left\lfloor x\right\rfloor\right)$.

Assume that $a,m$, and $n$ are integers such that $n>a$, $m>0$, and $f^{(2m)}(x)$ is absolutely integrable over $[a,n]$. Then

 2.10.1 $\sum_{j=a}^{n}f(j)=\int_{a}^{n}f(x)\,\mathrm{d}x+\tfrac{1}{2}f(a)+\tfrac{1}{2}% f(n)+\sum_{s=1}^{m-1}\frac{B_{2s}}{(2s)!}\left(f^{(2s-1)}(n)-f^{(2s-1)}(a)% \right)+\int_{a}^{n}\frac{B_{2m}-\widetilde{B}_{2m}\left(x\right)}{(2m)!}f^{(2% m)}(x)\,\mathrm{d}x.$

This is the Euler–Maclaurin formula. Another version is the Abel–Plana formula:

 2.10.2 $\sum_{j=a}^{n}f(j)=\int_{a}^{n}f(x)\,\mathrm{d}x+\tfrac{1}{2}f(a)+\tfrac{1}{2}% f(n)-2\int_{0}^{\infty}\frac{\Im\left(f(a+iy)\right)}{e^{2\pi y}-1}\,\mathrm{d% }y+\sum_{s=1}^{m}\frac{B_{2s}}{(2s)!}f^{(2s-1)}(n)+2\frac{(-1)^{m}}{(2m)!}\int% _{0}^{\infty}\Im\left(f^{(2m)}(n+i\vartheta_{n}y)\right)\frac{y^{2m}\,\mathrm{% d}y}{e^{2\pi y}-1},$

$\vartheta_{n}$ being some number in the interval $(0,1)$. Sufficient conditions for the validity of this second result are:

1. (a)

On the strip $a\leq\Re z\leq n$, $f(z)$ is analytic in its interior, $f^{(2m)}(z)$ is continuous on its closure, and $f(z)=o\left(e^{2\pi|\Im z|}\right)$ as $\Im z\to\pm\infty$, uniformly with respect to $\Re z\in[a,n]$.

2. (b)

$f(z)$ is real when $a\leq z\leq n$.

3. (c)

The first infinite integral in (2.10.2) converges.

### Example

 2.10.3 $S(n)=\sum_{j=1}^{n}j\ln j$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $n$: integer and $S(n)$: sum Permalink: http://dlmf.nist.gov/2.10.E3 Encodings: TeX, pMML, png See also: Annotations for §2.10(i), §2.10(i), §2.10 and Ch.2

for large $n$. From (2.10.1)

 2.10.4 $S(n)=\tfrac{1}{2}n^{2}\ln n-\tfrac{1}{4}n^{2}+\tfrac{1}{2}n\ln n+\tfrac{1}{12}% \ln n+C+\sum_{s=2}^{m-1}\frac{(-B_{2s})}{2s(2s-1)(2s-2)}\frac{1}{n^{2s-2}}+R_{% m}(n),$

where $m$ ($\geq 2$) is arbitrary, $C$ is a constant, and

 2.10.5 $R_{m}(n)=\int_{n}^{\infty}\frac{\widetilde{B}_{2m}\left(x\right)-B_{2m}}{2m(2m% -1)x^{2m-1}}\,\mathrm{d}x.$

From §24.12(i), (24.2.2), and (24.4.27), $\widetilde{B}_{2m}\left(x\right)-B_{2m}$ is of constant sign $(-1)^{m}$. Thus $R_{m}(n)$ and $R_{m+1}(n)$ are of opposite signs, and since their difference is the term corresponding to $s=m$ in (2.10.4), $R_{m}(n)$ is bounded in absolute value by this term and has the same sign.

Formula (2.10.2) is useful for evaluating the constant term in expansions obtained from (2.10.1). In the present example it leads to

 2.10.6 $C=\frac{\gamma+\ln\left(2\pi\right)}{12}-\frac{\zeta'\left(2\right)}{2\pi^{2}}% =\frac{1}{12}-\zeta'\left(-1\right),$

where $\gamma$ is Euler’s constant (§5.2(ii)) and $\zeta'$ is the derivative of the Riemann zeta function (§25.2(i)). $e^{C}$ is sometimes called Glaisher’s constant. For further information on $C$ see §5.17.

Other examples that can be verified in a similar way are:

 2.10.7 $\sum_{j=1}^{n-1}j^{\alpha}\sim\zeta\left(-\alpha\right)+\frac{n^{\alpha+1}}{% \alpha+1}\sum_{s=0}^{\infty}\genfrac{(}{)}{0.0pt}{}{\alpha+1}{s}\frac{B_{s}}{n% ^{s}},$ $n\to\infty$,

where $\alpha$ ($\neq-1$) is a real constant, and

 2.10.8 $\sum_{j=1}^{n-1}\frac{1}{j}\sim\ln n+\gamma-\frac{1}{2n}-\sum_{s=1}^{\infty}% \frac{B_{2s}}{2s}\frac{1}{n^{2s}},$ $n\to\infty$. ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\gamma$: Euler’s constant, $\sim$: Poincaré asymptotic expansion, $\ln\NVar{z}$: principal branch of logarithm function and $n$: integer Referenced by: §2.10(ii) Permalink: http://dlmf.nist.gov/2.10.E8 Encodings: TeX, pMML, png See also: Annotations for §2.10(i), §2.10(i), §2.10 and Ch.2

In both expansions the remainder term is bounded in absolute value by the first neglected term in the sum, and has the same sign, provided that in the case of (2.10.7), truncation takes place at $s=2m-1$, where $m$ is any positive integer satisfying $m\geq\frac{1}{2}(\alpha+1)$.

For extensions of the Euler–Maclaurin formula to functions $f(x)$ with singularities at $x=a$ or $x=n$ (or both) see Sidi (2004, 2012b, 2012a). See also Weniger (2007).

For an extension to integrals with Cauchy principal values see Elliott (1998).

## §2.10(ii) Summation by Parts

The formula for summation by parts is

 2.10.9 $\sum_{j=1}^{n-1}u_{j}v_{j}=U_{n-1}v_{n}+\sum_{j=1}^{n-1}U_{j}(v_{j}-v_{j+1}),$ ⓘ Symbols: $U_{j}$: coefficients, $u_{j}$: terms and $v_{j}$: terms Permalink: http://dlmf.nist.gov/2.10.E9 Encodings: TeX, pMML, png See also: Annotations for §2.10(ii), §2.10 and Ch.2

where

 2.10.10 $U_{j}=u_{1}+u_{2}+\dots+u_{j}.$ ⓘ Symbols: $U_{j}$: coefficients and $u_{j}$: terms Permalink: http://dlmf.nist.gov/2.10.E10 Encodings: TeX, pMML, png See also: Annotations for §2.10(ii), §2.10 and Ch.2

This identity can be used to find asymptotic approximations for large $n$ when the factor $v_{j}$ changes slowly with $j$, and $u_{j}$ is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i).

### Example

 2.10.11 $S(\alpha,\beta,n)=\sum_{j=1}^{n-1}e^{ij\beta}j^{\alpha},$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $S(\alpha,\beta,n)$: sum Permalink: http://dlmf.nist.gov/2.10.E11 Encodings: TeX, pMML, png See also: Annotations for §2.10(ii), §2.10(ii), §2.10 and Ch.2

where $\alpha$ and $\beta$ are real constants with $e^{i\beta}\neq 1$.

As a first estimate for large $n$

 2.10.12 $|S(\alpha,\beta,n)|\leq\sum_{j=1}^{n-1}j^{\alpha}=O\left(1\right),\;O\left(\ln n% \right),\text{ or }O\left(n^{\alpha+1}\right),$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\ln\NVar{z}$: principal branch of logarithm function and $S(\alpha,\beta,n)$: sum Referenced by: §2.10(ii) Permalink: http://dlmf.nist.gov/2.10.E12 Encodings: TeX, pMML, png See also: Annotations for §2.10(ii), §2.10(ii), §2.10 and Ch.2

according as $\alpha<-1$, $\alpha=-1$, or $\alpha>-1;$ see (2.10.7), (2.10.8). With $u_{j}=e^{ij\beta}$, $v_{j}=j^{\alpha}$,

 2.10.13 $U_{j}=e^{i\beta}(e^{ij\beta}-1)/(e^{i\beta}-1),$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $U_{j}$: coefficients Permalink: http://dlmf.nist.gov/2.10.E13 Encodings: TeX, pMML, png See also: Annotations for §2.10(ii), §2.10(ii), §2.10 and Ch.2

and

 2.10.14 $S(\alpha,\beta,n)=\frac{e^{i\beta}}{e^{i\beta}-1}\left(e^{i(n-1)\beta}n^{% \alpha}-1+\sum_{j=1}^{n-1}e^{ij\beta}\left(j^{\alpha}-(j+1)^{\alpha}\right)% \right).$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $S(\alpha,\beta,n)$: sum Permalink: http://dlmf.nist.gov/2.10.E14 Encodings: TeX, pMML, png See also: Annotations for §2.10(ii), §2.10(ii), §2.10 and Ch.2

Since

 2.10.15 $j^{\alpha}-(j+1)^{\alpha}=-\alpha j^{\alpha-1}+\alpha(\alpha-1)O\left(j^{% \alpha-2}\right)$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding Permalink: http://dlmf.nist.gov/2.10.E15 Encodings: TeX, pMML, png See also: Annotations for §2.10(ii), §2.10(ii), §2.10 and Ch.2

for any real constant $\alpha$ and the set of all positive integers $j$, we derive

 2.10.16 $S(\alpha,\beta,n)=\frac{e^{i\beta}}{e^{i\beta}-1}\left(e^{i(n-1)\beta}n^{% \alpha}-\alpha S(\alpha-1,\beta,n)+O\left(n^{\alpha-1}\right)+O\left(1\right)% \right).$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $S(\alpha,\beta,n)$: sum Referenced by: §2.10(ii) Permalink: http://dlmf.nist.gov/2.10.E16 Encodings: TeX, pMML, png See also: Annotations for §2.10(ii), §2.10(ii), §2.10 and Ch.2

From this result and (2.10.12)

 2.10.17 $S(\alpha,\beta,n)=O\left(n^{\alpha}\right)+O\left(1\right).$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding and $S(\alpha,\beta,n)$: sum Permalink: http://dlmf.nist.gov/2.10.E17 Encodings: TeX, pMML, png See also: Annotations for §2.10(ii), §2.10(ii), §2.10 and Ch.2

Then replacing $\alpha$ by $\alpha-1$ and resubstituting in (2.10.16), we have

 2.10.18 $S(\alpha,\beta,n)=\frac{e^{in\beta}}{e^{i\beta}-1}n^{\alpha}+O\left(n^{\alpha-% 1}\right)+O\left(1\right),$ $n\to\infty$, ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $S(\alpha,\beta,n)$: sum Permalink: http://dlmf.nist.gov/2.10.E18 Encodings: TeX, pMML, png See also: Annotations for §2.10(ii), §2.10(ii), §2.10 and Ch.2

which is a useful approximation when $\alpha>0$.

For extensions to $\alpha\leq 0$, higher terms, and other examples, see Olver (1997b, Chapter 8).

## §2.10(iii) Asymptotic Expansions of Entire Functions

The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5.

### Example

From §§16.2(i)16.2(ii)

 2.10.19 ${{}_{0}F_{2}}\left(-;1,1;x\right)=\sum_{j=0}^{\infty}\frac{x^{j}}{(j!)^{3}}.$

We seek the behavior as $x\to+\infty$. From (1.10.8)

 2.10.20 $\sum_{j=0}^{n-1}\frac{x^{j}}{(j!)^{3}}=\frac{1}{2i}\int_{\mathscr{C}}\frac{x^{% t}}{(\Gamma\left(t+1\right))^{3}}\cot\left(\pi t\right)\,\mathrm{d}t,$

where $\mathscr{C}$ comprises the two semicircles and two parts of the imaginary axis depicted in Figure 2.10.1.

From the identities

 2.10.21 $\frac{\cot\left(\pi t\right)}{2i}=-\frac{1}{2}-\frac{1}{e^{-2\pi it}-1}=\frac{% 1}{2}+\frac{1}{e^{2\pi it}-1},$

and Cauchy’s theorem, we have

 2.10.22 $\sum_{j=0}^{n-1}\frac{x^{j}}{(j!)^{3}}=\int_{-1/2}^{n-(1/2)}\frac{x^{t}}{(% \Gamma\left(t+1\right))^{3}}\,\mathrm{d}t-\int_{\mathscr{C}_{1}}\frac{x^{t}}{(% \Gamma\left(t+1\right))^{3}}\frac{\,\mathrm{d}t}{e^{-2\pi it}-1}+\int_{% \mathscr{C}_{2}}\frac{x^{t}}{(\Gamma\left(t+1\right))^{3}}\frac{\,\mathrm{d}t}% {e^{2\pi it}-1},$

where $\mathscr{C}_{1},\mathscr{C}_{2}$ denote respectively the upper and lower halves of $\mathscr{C}$. (5.11.7) shows that the integrals around the large quarter circles vanish as $n\to\infty$. Hence

 2.10.23 ${{}_{0}F_{2}}\left(-;1,1;x\right)=\int_{-1/2}^{\infty}\frac{x^{t}}{(\Gamma% \left(t+1\right))^{3}}\,\mathrm{d}t+2\Re\int_{-1/2}^{i\infty}\frac{x^{t}}{(% \Gamma\left(t+1\right))^{3}}\frac{\,\mathrm{d}t}{e^{-2\pi it}-1}=\int_{0}^{% \infty}\frac{x^{t}}{(\Gamma\left(t+1\right))^{3}}\,\mathrm{d}t+O\left(1\right),$ $x\to+\infty$,

the last step following from $|x^{t}|\leq 1$ when $t$ is on the interval $[-\frac{1}{2},0]$, the imaginary axis, or the small semicircle. By application of Laplace’s method (§2.3(iii)) and use again of (5.11.7), we obtain

 2.10.24 ${{}_{0}F_{2}}\left(-;1,1;x\right)\sim\frac{\exp\left(3x^{1/3}\right)}{2\pi 3^{% 1/2}x^{1/3}},$ $x\to+\infty$.

For generalizations and other examples see Olver (1997b, Chapter 8), Ford (1960), and Berndt and Evans (1984). See also Paris and Kaminski (2001, Chapter 5) and §§16.11(i)16.11(ii).

## §2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method

Let $f(z)$ be analytic on the annulus $0<|z|, with Laurent expansion

 2.10.25 $f(z)=\sum_{n=-\infty}^{\infty}f_{n}z^{n},$ $0<|z|. ⓘ Symbols: $f(x)$: analytic function, $f_{n}$: coefficients and $r$: radious of annulus Permalink: http://dlmf.nist.gov/2.10.E25 Encodings: TeX, pMML, png See also: Annotations for §2.10(iv), §2.10 and Ch.2

What is the asymptotic behavior of $f_{n}$ as $n\to\infty$ or $n\to-\infty$? More specially, what is the behavior of the higher coefficients in a Taylor-series expansion?

These problems can be brought within the scope of §2.4 by means of Cauchy’s integral formula

 2.10.26 $f_{n}=\frac{1}{2\pi i}\int_{\mathscr{C}}\frac{f(z)}{z^{n+1}}\,\mathrm{d}z,$

where $\mathscr{C}$ is a simple closed contour in the annulus that encloses $z=0$. For examples see Olver (1997b, Chapters 8, 9).

However, if $r$ is finite and $f(z)$ has algebraic or logarithmic singularities on $|z|=r$, then Darboux’s method is usually easier to apply. We need a “comparison function” $g(z)$ with the properties:

1. (a)

$g(z)$ is analytic on $0<|z|.

2. (b)

$f(z)-g(z)$ is continuous on $0<|z|\leq r$.

3. (c)

The coefficients in the Laurent expansion

 2.10.27 $g(z)=\sum_{n=-\infty}^{\infty}g_{n}z^{n},$ $0<|z|, ⓘ Symbols: $r$: radious of annulus, $g(z)$: comparison function and $g_{n}$: coefficients Permalink: http://dlmf.nist.gov/2.10.E27 Encodings: TeX, pMML, png See also: Annotations for §2.10(iv), §2.10 and Ch.2

have known asymptotic behavior as $n\to\pm\infty$.

By allowing the contour in Cauchy’s formula to expand, we find that

 2.10.28 $f_{n}-g_{n}=\frac{1}{2\pi i}\int_{|z|=r}\frac{f(z)-g(z)}{z^{n+1}}\,\mathrm{d}z% =\frac{1}{2\pi r^{n}}\int_{0}^{2\pi}\left(f\left(re^{i\theta}\right)-g\left(re% ^{i\theta}\right)\right)e^{-ni\theta}\,\mathrm{d}\theta.$

Hence by the Riemann–Lebesgue lemma (§1.8(i))

 2.10.29 $f_{n}=g_{n}+o\left(r^{-n}\right),$ $n\to\pm\infty$. ⓘ Symbols: $o\left(\NVar{x}\right)$: order less than, $f_{n}$: coefficients, $r$: radious of annulus and $g_{n}$: coefficients Referenced by: §2.10(iv) Permalink: http://dlmf.nist.gov/2.10.E29 Encodings: TeX, pMML, png See also: Annotations for §2.10(iv), §2.10 and Ch.2

This result is refinable in two important ways. First, the conditions can be weakened. It is unnecessary for $f(z)-g(z)$ to be continuous on $|z|=r$: it suffices that the integrals in (2.10.28) converge uniformly. For example, Condition (b) can be replaced by:

1. (b´)

On the circle $|z|=r$, the function $f(z)-g(z)$ has a finite number of singularities, and at each singularity $z_{j}$, say,

 2.10.30 $f(z)-g(z)=O\left((z-z_{j})^{\sigma_{j}-1}\right),$ $z\to z_{j}$, ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $f(x)$: analytic function, $g(z)$: comparison function and $\sigma_{j}$: positive constant Permalink: http://dlmf.nist.gov/2.10.E30 Encodings: TeX, pMML, png See also: Annotations for §2.10(iv), §2.10 and Ch.2

where $\sigma_{j}$ is a positive constant.

Secondly, when $f(z)-g(z)$ is $m$ times continuously differentiable on $|z|=r$ the result (2.10.29) can be strengthened. In these circumstances the integrals in (2.10.28) are integrable by parts $m$ times, yielding

 2.10.31 $f_{n}=g_{n}+o\left(r^{-n}|n|^{-m}\right),$ $n\to\pm\infty$. ⓘ Symbols: $o\left(\NVar{x}\right)$: order less than, $f_{n}$: coefficients, $r$: radious of annulus and $g_{n}$: coefficients Referenced by: §2.10(iv), §2.10(iv) Permalink: http://dlmf.nist.gov/2.10.E31 Encodings: TeX, pMML, png See also: Annotations for §2.10(iv), §2.10 and Ch.2

Furthermore, (2.10.31) remains valid with the weaker condition

 2.10.32 $f^{(m)}(z)-g^{(m)}(z)=O\left((z-z_{j})^{\sigma_{j}-1}\right),$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $f(x)$: analytic function, $g(z)$: comparison function and $\sigma_{j}$: positive constant Permalink: http://dlmf.nist.gov/2.10.E32 Encodings: TeX, pMML, png See also: Annotations for §2.10(iv), §2.10 and Ch.2

in the neighborhood of each singularity $z_{j}$, again with $\sigma_{j}>0$.

### Example

Let $\alpha$ be a constant in $(0,2\pi)$ and $P_{n}$ denote the Legendre polynomial of degree $n$. From §14.7(iv)

 2.10.33 $f(z)\equiv\frac{1}{(1-2z\cos\alpha+z^{2})^{1/2}}=\sum_{n=0}^{\infty}P_{n}\left% (\cos\alpha\right)z^{n},$ $|z|<1$.

The singularities of $f(z)$ on the unit circle are branch points at $z=e^{\pm i\alpha}$. To match the limiting behavior of $f(z)$ at these points we set

 2.10.34 $g(z)=e^{-\pi i/4}(2\sin\alpha)^{-1/2}\left(e^{-i\alpha}-z\right)^{-1/2}+e^{\pi i% /4}(2\sin\alpha)^{-1/2}\left(e^{i\alpha}-z\right)^{-1/2}.$

Here the branch of $\left(e^{-i\alpha}-z\right)^{-1/2}$ is continuous in the $z$-plane cut along the outward-drawn ray through $z=e^{-i\alpha}$ and equals $e^{i\alpha/2}$ at $z=0$. Similarly for $\left(e^{i\alpha}-z\right)^{-1/2}$. In Condition (c) we have

 2.10.35 $g_{n}=\left(\frac{2}{\pi\sin\alpha}\right)^{1/2}\frac{\Gamma\left(n+\frac{1}{2% }\right)}{n!}\cos\left(n\alpha+\tfrac{1}{2}\alpha-\tfrac{1}{4}\pi\right),$

and in the supplementary conditions we may set $m=1$. Then from (2.10.31) and (5.11.7)

 2.10.36 $P_{n}\left(\cos\alpha\right)=\left(\frac{2}{\pi n\sin\alpha}\right)^{1/2}\cos% \left(n\alpha+\tfrac{1}{2}\alpha-\tfrac{1}{4}\pi\right)+o\left(n^{-1}\right).$

For higher terms see §18.15(iii).

For uniform expansions when two singularities coalesce on the circle of convergence see Wong and Zhao (2005).

For other examples and extensions see Olver (1997b, Chapter 8), Olver (1970), Wong (1989, Chapter 2), and Wong and Wyman (1974). See also Flajolet and Odlyzko (1990).