---
title: 'Bateman-Type Sum: Special Functions and Number Theory'
url: https://www.emergentmind.com/topics/bateman-type-sum
type: topic
---

# Bateman-Type Sum: Special Functions and Number Theory

Searching arXiv for the cited paper and closely related Bateman-function survey to ground the article in current sources.
A Bateman-type sum designates, in the cited arXiv literature, two distinct classes of objects. In the special-function setting, it refers to explicit summation formulas built from the Bateman functions \(k_\nu(x)\) and the allied Havelock functions \(h_\nu(x)\), typically obtained from generating functions, recurrences, and integral representations [2104.08596]. In analytic number theory, the phrase also appears for logarithmic sums associated with the tail of the Bateman–Horn singular product, where one studies expressions such as \(\sum_{p>P}\ln(1-\nu_p/p)\) and the corresponding prime-product tail \(T(P)\) for one-dimensional systems of polynomials [2606.28832]. The common feature is the reduction of a structured infinite product or family of special functions to a controlled summation formula.

## 1. Special-function framework

The Bateman functions are defined, for any real order \(\nu\), by the finite integrals
\[
k_{\nu}(x)=\frac{2}{\pi}\int_0^{\pi/2}\cos(x\tan\theta-\nu\theta)\,d\theta,
\qquad
h_{\nu}(x)=\frac{2}{\pi}\int_0^{\pi/2}\sin(x\tan\theta-\nu\theta)\,d\theta.
\]
They are related to Whittaker, Tricomi, and Kummer functions through
\[
k_{2\nu}(x)=\frac{1}{\Gamma(\nu+1)}W_{\nu,1/2}(2x)
= \frac{e^{-x}}{\Gamma(\nu+1)}U(-\nu,0;2x),
\]
with
\[
U(-\nu,0;2x)=2x\,U(1-\nu,2;2x).
\]
For integer \(n\ge 0\),
\[
k_{2n+2}(x)=2x\,e^{-x}\,{}_1F_1(-2n;2;2x).
\]
These identities place Bateman functions within the confluent-hypergeometric hierarchy and provide the standard analytic entry point for closed-form Bateman-type sums [2104.08596].

A corresponding power-series expansion follows from the defining series of \({}_1F_1\):
\[
k_{2n+2}(x)
=
2x\,e^{-x}\sum_{m=0}^{2n}
\frac{(-2n)_m}{(2)_m}\frac{(2x)^m}{m!}.
\]
By analytic continuation one may write, for general \(\nu\),
\[
k_{2\nu}(x)=\frac{e^{-x}}{\Gamma(\nu+1)}\sum_{m=0}^\infty
\frac{(-\nu)_m}{(1)_m}\frac{(2x)^m}{m!}.
\]
This suggests that Bateman-type sums in the special-function sense are naturally organized by hypergeometric truncation for integer orders and by analytic continuation for general order.

## 2. Canonical Bateman-type sums

The most direct Bateman-type sums arise from generating functions and admit exact closed forms. A central identity is the exponential-kernel sum
\[
\sum_{n=0}^\infty (-1)^n t^n\,k_{2n}(x)
=
\exp\!\Bigl[-\,\frac{x(1+t)}{1-t}\Bigr],
\qquad |t|<1.
\]
In the survey, this is presented as the exponential generating function for even Bateman orders, with \(F_n(x)\coloneqq (-1)^n k_{2n}(x)\) [2104.08596].

Two particularly compact evaluations are
\[
\sum_{n=0}^\infty (-1)^n\,k_{4n}(x)=\cos x,
\qquad
\sum_{n=0}^\infty (-1)^n\,k_{4n+2}(x)=\sin x.
\]
These formulas exhibit a direct bridge between Bateman-function series and elementary trigonometric functions. A further polynomially weighted identity is
\[
\sum_{n=0}^\infty (n+1)\,t^n\,k_{2n+2}(x)
=
\frac{2x}{(1+t)^2}\,e^{-x+\tfrac{2xt}{1+t}}.
\]

The survey also records trigonometric-kernel sums involving the Havelock and Bateman functions:
\[
\sum_{n=1}^\infty t^n\,h_n(nx)
=
\frac{2}{\pi}\int_0^{\pi/2}
\frac{t\sin(x\tan\theta-\theta)}
{1-2t\cos(x\tan\theta-\theta)+t^2}\,d\theta,
\]
\[
\sum_{n=1}^\infty t^n\,k_n(nx)
=
\frac{2}{\pi}\int_0^{\pi/2}
\frac{1-t\cos(x\tan\theta-\theta)}
{1-2t\cos(x\tan\theta-\theta)+t^2}\,d\theta,
\qquad |t|<1.
\]
In addition, a mixed Bateman–Havelock identity is given, for even \(n\), by
\[
(n-2)\bigl[k_n h_{n-2}-k_{n-2}h_n\bigr]
+
(n+2)\bigl[k_n h_{n+2}-k_{n+2}h_n\bigr]
=
-\frac{8}{\pi}\,k_n.
\]

The domain stated for these prototype Bateman-type sums is \(x>0\) and \(|t|<1\) unless otherwise specified. The survey notes applications to plane-wave problems in fluid mechanics, summation of Laguerre-type polynomials weighted by exponential factors, and evaluation of infinite integrals in potential-flow and wave-resistance theory.

## 3. Structural identities underlying the sums

The Bateman-type sums in the special-function literature are supported by a system of recurrence, differential-difference, symmetry, and transform identities. For \(n\in\mathbb Z\) and \(x>0\), the Bateman functions satisfy
\[
(2x-2n)\,k_{2n}(x)=(n-1)\,k_{2n-2}(x)+(n+1)\,k_{2n+2}(x),
\]
\[
4x\,k'_n(x)=(n-2)\,k_{n-2}(x)-(n+2)\,k_{n+2}(x),
\]
\[
k'_n(x)+k'_{n+2}(x)=k_n(x)-k_{n+2}(x),
\]
\[
x\,k''_n(x)=(x-n)\,k_n(x).
\]
A mirror symmetry is also recorded:
\[
k_{-n}(x)=k_n(-x).
\]
Analogous recurrences hold for \(h_n(x)\), including
\[
(2n-4x)\,h_n+(n-2)\,h_{n-2}+(n+2)\,h_{n+2}=-\frac{8}{\pi}.
\]
These formulas are the algebraic machinery from which further Bateman-type sums can be derived [2104.08596].

The same structural role is played by generating functions and Laplace transforms. Besides
\[
\sum_{n=0}^\infty (-1)^n t^n k_{2n}(x)
=
\exp\!\Bigl[-\,\frac{x(1+t)}{1-t}\Bigr],
\]
the survey gives
\[
\sum_{n=0}^\infty \frac{t^n}{2^n n!}\,k_{2n+2}(x)
=
2\,e^{-(x+t/2)}\sqrt{\frac{x}{t}\,I_1(2\sqrt{x\,t})},
\]
together with Laplace transforms
\[
\mathcal L\{k_0(t)\}(s)=\frac{1}{s+1},
\qquad
\mathcal L\{k_{2n+2}(t)\}(s)=\frac{2(1-s)^n}{(s+1)^{n+2}},
\qquad \Re s>-1.
\]
The survey explicitly states that such identities constitute a toolbox for constructing further Bateman-type sums by analytic continuation in \(\nu\), by combining recurrence relations with generating functions, or by applying Laplace or Mellin transforms to obtain hybrid summation–integral formulas. This suggests that the term is best understood as a family of structurally related closed-form summations rather than a single canonical series.

## 4. Arithmetic Bateman-type sums and the singular product

In the arithmetic setting, let \(\mathbf f=(f_1,\dots,f_k)\) be a one-dimensional system of integer polynomials, each irreducible and without multiple roots. For each prime \(p\), define
\[
\nu_p(\mathbf f)
=
\#\{\,n \pmod p:\; p\mid f_1(n)\cdots f_k(n)\,\}.
\]
The local singular factor may be written as
\[
L_p=\frac{1-\nu_p(\mathbf f)/p}{(1-1/p)^k},
\]
and the singular product is identified in the exposition with
\[
\mathfrak S(\mathbf f)\equiv \prod_p\Bigl(1-\frac{\nu_p(\mathbf f)}{p}\Bigr).
\]
The tail starting at a cutoff \(P\) is
\[
T(P)=\prod_{p>P}\Bigl(1-\frac{\nu_p}{p}\Bigr).
\]
In this context, the paper isolates a “Bateman-type sum via logarithms” by expanding the logarithm of the tail [2606.28832].

The universal bound proved in Theorem 3.1 is
\[
\ln T(P)=O\!\bigl(1/\ln P\bigr),
\qquad P\to\infty,
\]
and hence
\[
\prod_{p>P}\Bigl(1-\frac{\nu_p}{p}\Bigr)=1+O\!\bigl(1/\ln P\bigr).
\]
The logarithmic expansion is
\[
\ln\Bigl(1-\frac{\nu_p}{p}\Bigr)
=
-\frac{\nu_p}{p}+O\!\bigl(p^{-2}\bigr),
\]
which yields
\[
\sum_{p>P}\ln\Bigl(1-\frac{\nu_p}{p}\Bigr)
=
-\sum_{p>P}\frac{\nu_p}{p}
+
O\!\bigl(P^{-1}\bigr).
\]
The exposition states that \(O(P^{-1})\) may be refined to \(O(1/\ln P)\) via Chebotarev. In this usage, the Bateman-type sum is not a special-function series but a logarithmic prime sum governing the convergence of the singular product.

## 5. Linear, nonlinear, and mixed systems

For linear systems, all \(f_i\) are linear forms, their splitting field is \(\mathbb Q\), and the Galois group is trivial. The paper shows that for all sufficiently large \(p\), \(\nu_p=k\), so that
\[
\ln\Bigl(1-\frac{k}{p}\Bigr)
=
-\frac{k}{p}-\frac{k^2}{2p^2}+O(p^{-3})
=
-\frac{k}{p}-\frac{k(k-1)}{2p^2}+O(p^{-3}).
\]
Summing over \(p>P\) and using \(\sum_{p>P}p^{-2}\asymp 1/(P\ln P)\), one obtains the exact asymptotic
\[
\ln T(P)
=
-\frac{k(k-1)}{2}\,\frac{1}{P\ln P}
+
O\!\bigl(\tfrac{1}{P\ln^2 P}\bigr),
\]
and therefore
\[
T(P)
=
1-\frac{k(k-1)}{2}\,\frac{1}{P\ln P}
+
O\!\bigl(\tfrac{1}{P\ln^2 P}\bigr).
\]
The exposition emphasizes that no unproven hypothesis is needed in the linear case [2606.28832].

For nonlinear systems, let \(K\) be the splitting field of \(F=f_1\cdots f_k\) and \(G=\mathrm{Gal}(K/\mathbb Q)\). If \(\mathrm{Fix}(g)\) denotes the number of geometric roots of \(F\) fixed by \(g\in G\), the coefficient
\[
C(F)=\frac{1}{|G|}\sum_{g\in G}\bigl(k-\mathrm{Fix}(g)\bigr)
\]
is introduced. Equivalently, via Chebotarev, \(C(F)\) is the mean value of \(k-\nu_p\) as \(p\to\infty\). If \(G\) is abelian and \(C(F)=0\), then under the Generalized Riemann Hypothesis for all Dirichlet \(L(s,\chi)\) attached to the characters of \(G\),
\[
\ln T(P)=O\!\bigl(P^{-1/2+\varepsilon}\bigr)
\qquad(\forall\,\varepsilon>0),
\]
and hence
\[
T(P)=1+O\!\bigl(P^{-1/2+\varepsilon}\bigr).
\]

For mixed systems, \(F\) contains a subset \(C\) of linear forms and a subset \(N\) of nonlinear irreducibles. In the abelian or trivial-intersection case,
\[
C(F)=k-m,
\]
where \(k=|C|+|N|\) is the total number of polynomials and \(m\) the number of distinct irreducible factors over \(\mathbb Q\). The tail splits multiplicatively,
\[
T(P)=T_{\rm lin}(P)\times T_{\rm nonlin}(P),
\]
so that
\[
\ln T(P)
=
-\frac{|C|(|C|-1)}{2P\ln P}
+
O\!\bigl(P^{-1/2+\varepsilon}\bigr)
\]
under GRH, or \(O(1/\ln P)\) unconditionally. This establishes a precise separation between the superfast linear contribution and the slower nonlinear contribution.

## 6. Numerical behavior and conceptual scope

The numerical tables in the tail-estimate paper compare the partial products
\[
G(X)=\prod_{p\le X}\Bigl(1-\frac{\nu_p}{p}\Bigr)
\]
with the conjectural limit \(\mathfrak S(\mathbf f)\), using the relative error
\[
\frac{|G(X)-\mathfrak S(\mathbf f)|}{\mathfrak S(\mathbf f)}.
\]
The reported examples are as follows [2606.28832].

| System \(F\) | \(C(F)\) | Reported error and convergence |
|---|---:|---|
| \(n^2+1\) | \(<0\) | \(<0.5\%\) at \(10^4\); \(<0.1\%\) at \(10^6\); fast \((O(X^{-1/2+\varepsilon}),\ \mathrm{GRH})\) |
| \(n^4+3n^2+1\) | \(<0\) | \(<0.5\%\) at \(10^4\); \(<0.1\%\) at \(10^6\); fast (GRH) |
| \(n^2+1,\;n^2+2\) | \(<0\) | \(<0.5\%\) at \(10^4\); \(<0.1\%\) at \(10^6\); fast (GRH) |
| \(n^2+1,\;n^2+2,\;n^2+3\) | \(<0\) | \(<0.5\%\) at \(10^4\); \(<0.1\%\) at \(10^6\); fast (GRH) |
| Twin primes \((0,2)\), \(k=2\) | \(0\) | \(\sim 0.01\%\) at \(10^4\); \(0.0001\%\) at \(10^6\); superfast \((O(1/(X\ln X)))\) |
| Prime triplet \((0,2,6)\), \(k=3\) | \(0\) | \(\sim 0.01\%\) at \(10^4\); \(0.0001\%\) at \(10^6\); superfast |
| Goldbach pair \(\{n,N-n\}\), \(k=2\) | \(0\) | \(\sim 0.01\%\) at \(10^4\); \(0.0001\%\) at \(10^6\); superfast |
| Mixed \(n,\;n^2+1\) | \(0\) | \(<0.5\%\) at \(10^4\); \(<0.1\%\) at \(10^6\); fast (GRH) |

The exposition states that in all abelian-nonlinear examples GRH is assumed to obtain the fast power saving, whereas without GRH one still has the universal \(O(1/\ln X)\) bound, which at \(X=10^6\) would correspond to a 10\%-level error. The linear examples, by contrast, match the unconditional superfast \(1/(X\ln X)\) rate.

A common source of confusion is terminological rather than mathematical. In the cited sources, “Bateman-type sum” does not denote a single universally standardized object. One usage belongs to the analysis of Bateman and Havelock functions, where the emphasis is on generating-function and transform identities; the other belongs to Bateman–Horn theory, where the emphasis is on logarithmic expansions of prime-product tails. The available arXiv evidence therefore supports a two-branch understanding: Bateman-type sums are either exact summation formulas for Bateman-function families or asymptotic logarithmic sums attached to singular products. This suggests that the unifying principle is formal summation structure, not a shared analytic domain.

Source: https://www.emergentmind.com/topics/bateman-type-sum