# Analysis of nonresonant effects in the two-photon spectroscopy of helium

T. Zalialiutdinov1, A. Anikin1, D. Solovyev1 1 Department of Physics, St.
Petersburg State University, Petrodvorets, Oulianovskaya 1, 198504, St.
Petersburg, Russia

###### Abstract

In the present paper, we study nonresonant corrections for experimental
measurements of the transition frequencies in the helium atom. Having
attracted more attention, such effects can make a significant contribution to
experiments based on one- and two-photon atomic spectroscopy. The quantum
interference effects in the measurements of $n^{3}S_{1}-3^{3}D_{1}$
($n=3,\,4,\,5$) transition frequencies based on Doppler-free two-photon
spectroscopy, are considered as a possible source of current discrepancy
between the experimental and theoretical data. We demonstrate that line
profile asymmetry caused by the quantum interference of fine sub-levels of the
${}^{3}D_{J}$ $(J=1,2,3)$ state can reach tenths of a megahertz for different
experimental conditions. Thus, previously unaccounted nonresonant corrections
should be taken in next-generation experimental measurements of transitions
frequencies in helium. However, they could not completely eliminate the
current imbalance in the study of helium spectra and the question is still
open.

Over the past two decades, the absolute measurement of transition frequencies
in simple atomic systems like hydrogen and helium has been continuously
improved with the aim of determining the fundamental physical constants,
nuclear properties, testing quantum electrodynamics calculations. Recent
precise measurements of transition frequencies in hydrogen have achieved a
level where the quantum interference effects (QIE) between closest degenerate
states have played a decisive role Beyer and et al. (2017); Yost  _et al._
(2014). Significant progress in the spectroscopy of one-electron systems has
spurred on the study of nonresonant (NR) corrections to the transition
energies in many-electron systems Amaro  _et al._ (2015); Brown  _et al._
(2013); Marsman  _et al._ (2015); Sansonetti  _et al._ (2011). Although helium
has been studied theoretically and experimentally for many years, accounting
for the nonresonant effect Jentschura and Mohr (2002) and the QIE as a part of
them in spectroscopic measurements of the transition frequencies has not been
considered until recently Amaro  _et al._ (2015); Marsman  _et al._ (2015).

The energies of atomic levels in helium are conventionally expressed as the
sum of nonrelativistic energies, lowest-order relativistic corrections, Lamb
shift, etc., which includes quantum electrodynamics corrections (QED) and
higher-order relativistic terms. Recent calculations of QED effects at the
$\alpha^{7}m$ level have improved theoretical predictions for the energies of
helium atomic levels, leading to complete agreement with the measured
$2^{3}S-2^{3}P$ transition frequency Patkóš  _et al._ (2021). However, as
found in Patkóš  _et al._ (2021), such calculations do not eliminate the
discrepancy between theoretical predictions and the experimental result for
the $2^{3}S_{1}-3^{3}D_{1}$ transition discussed earlier in Yerokhin  _et al._
(2020).

This work is devoted to a theoretical analysis of the nonresonant effects
arising from neighbouring fine structure sublevels, when measuring the energy
of the $2^{3}S_{1}-3^{3}D_{1}$ transition, and is aimed at eliminating the
emerged theoretical and experimental discrepancies by taking into account the
QIE correction. In the experiment reported in Dorrer  _et al._ (1997), helium
atoms in an atomic beam are prepared in metastable $2^{3}S_{1}$ state and then
excited into the $3^{3}D_{1}$ state by absorbing two photons with equal
frequencies, $\omega_{1}=\omega_{2}$ having parallel polarizations
$\textbf{e}_{1}$ and $\textbf{e}_{2}$, and propagating in opposite directions
Schwob  _et al._ (1999). The detection of an excited fraction of $3^{3}D_{1}$
atoms is observed by fluorescence (i.e., decay into $2^{3}P$ states) with the
emission of a photon with a frequency $\omega_{3}$, polarization
$\textbf{e}_{3}^{*}$ towards direction $\bm{\nu}_{3}$. In this case, there
should be interference between sublevels of the fine structure $3^{3}D_{1}$,
$3^{3}D_{2}$ and $3^{3}D_{3}$, which leads to an asymmetry of the line profile
Labzowsky  _et al._ (2001). This experimental situation is similar to that
previously considered in Anikin  _et al._ (2021), where NR corrections to the
energies of $2s-ns/nd$ transitions in hydrogen ($n=4,\;6,\;8,\;12$ is the
principal quantum number) with interference between $nd^{F=2}_{3/2}$ and
$nd^{F=2}_{5/2}$ states were studied. Applying the QED theory of the spectral
line profile, see Andreev  _et al._ (2008), we describe the NR correction to
the transition frequency $2^{3}S_{1}-3^{3}D_{1}$ for the experimental scheme
given in Dorrer  _et al._ (1997).

Following Dorrer  _et al._ (1997) we are interested in the case when two
incident photons with polarizations $\textbf{e}_{1}$ and $\textbf{e}_{2}$
propagating in opposite directions given by vectors $\bm{\nu}_{1}$ and
$\bm{\nu}_{2}$, respectively, are absorbed from initially prepared
$2^{3}S_{1}$ state into the intermediate state $3^{3}D_{J_{n}}$, thereat
$\omega_{1}+\omega_{2}=E_{3^{3}D_{J_{n}}}-E_{2^{3}S_{1}}$. Two-photon
selection rules allow the transition to the different fine sublevels with
$J_{n}=1,\,2,\,3$. Then the fluorescent signal in a direction $\bm{\nu}_{3}$
of photons with frequency $\omega_{3}$ and polarization $\textbf{e}_{3}^{*}$
emitted in electric dipole transition $3^{3}D_{J_{n}}\rightarrow
2^{3}P_{J_{f}}+\gamma(\mathrm{E1})$ is observed. Performing summation over
projections of angular momenta of final and intermediate states, averaging
over projections of initial state, integrating over the frequency $\omega_{3}$
the corresponding cross section of scattering process can be written as Anikin
_et al._ (2021)

$\displaystyle\sigma(\omega_{1},\omega_{2})=\frac{\mathrm{const}}{2J_{i}+1}\sum\limits_{J_{f}}\sum\limits_{M_{J_{i}}M_{J_{f}}}\left|\sum\limits_{\begin{subarray}{c}J_{n}M_{J_{n}}\\\
kJ_{k}M_{J_{k}}\end{subarray}}(E_{2^{3}S_{1}}-E_{2^{3}P_{J_{f}}}+\omega_{1}+\omega_{2})^{3/2}\omega_{2}^{1/2}\omega_{1}^{1/2}\frac{\langle
2^{3}P_{J_{f}}|\textbf{e}_{3}^{*}\textbf{D}|3^{3}D_{J_{n}}\rangle}{E_{3^{3}D_{J_{n}}}-E_{2^{3}S_{1}}-\omega_{1}-\omega_{2}}\right.$
(1) $\displaystyle\left.\times\frac{\langle
3^{3}D_{J_{n}}|\textbf{e}_{2}\textbf{D}|k^{3}P_{J_{k}}\rangle\langle
k^{3}P_{J_{k}}|\textbf{e}_{1}\textbf{D}|2^{3}S_{1}\rangle}{E_{k^{3}P_{J_{k}}}-E_{2^{3}S_{1}}-\omega_{2}}+\mbox{5
permutations}\right|^{2},$

where $\textbf{D}=e(\textbf{r}_{1}+\textbf{r}_{2})$ is the electric dipole
operator of two-electron system ($e$ is the charge of electron) and
’$\mathrm{const}$’ is a some numerical factor which is unimportant for further
calculations. Permutations in Eq. (1) are understood as all possible
rearrangement of indices 1, 2, 3 denoting the corresponding photons. Since the
detector in the experiment Dorrer  _et al._ (1997) do not resolve energies of
fluorescent photon connected with the decay to different fine sublevels
$J_{f}=0,\,1,\,2$ an additional summation over $J_{f}$ appeared in Eq. (1).
Sum over $k$ in Eq. (1) implies summation over the entire spectrum of the
Schrödinger equation for two-electron system including integration over the
continuous spectrum. The calculation of total cross section for two- and
three-photon scattering processes in helium with summation over the spectrum
of fully correlated two-electron intermediate states was considered earlier in
works Drake  _et al._ (1969); Zalialiutdinov  _et al._ (2016). In Eq. (1) we
also neglect the contribution of higher-order multipoles, since their
contribution to the NR effects is negligible at the current experimental
accuracy Labzowsky  _et al._ (1993).

The matrix elements of Eq. (1) depend on the photon directions $\bm{\nu}_{1}$,
$\bm{\nu}_{2}$ and $\bm{\nu}_{3}$ through the transversality condition of the
polarization vectors. In the Dorrer  _et al._ (1997) experiment, incident
photons propagate in opposite directions with polarization vectors
$\textbf{e}_{1}$ and $\textbf{e}_{2}$, and the outgoing photon has
polarization $\textbf{e}_{3}^{*}$, whose directions are considered fixed.
Then, denoting the angles between any pair of two vectors as $\theta_{ij}$
($i,j=1,\;2,\;3$), we will be interested in the arsing angular correlations
depending on these angles when determining the transition frequency from the
cross section (1).

Since the experiment Dorrer  _et al._ (1997) is aimed at measuring the two-
photon absorption transition $2^{3}S_{1}+2\gamma(\mathrm{E1})\rightarrow
3^{3}D_{1}$, we employ the resonant approximation Andreev  _et al._ (2008).
Within the framework of this simplification, the nonresonant terms describing
transitions to other levels $3^{3}D_{J_{n}=2,\,3}$ in the scattering amplitude
Eq. (1) can be omitted with high accuracy, what is justified only if the
corresponding NR corrections go beyond the accuracy of experiments Solovyev
_et al._ (2020a). According to Dorrer  _et al._ (1997), the frequencies of two
incident laser photons are equal

$\displaystyle\omega_{1}=\omega_{2}=\omega=(E_{3^{3}D_{1}}-E_{2^{3}S_{1}})/2,$

and the detector of fluorescent signal is insensitive to the polarization
$\textbf{e}_{3}^{*}$ of the outgoing photon. Then, taking into account that
$\omega_{1}+\omega_{2}=E_{3^{3}D_{1}}-E_{2^{3}S_{1}}$ and summing over the
polarization $\textbf{e}_{3}^{*}$ with the use of equation Labzowsky  _et al._
(1993)

$\displaystyle\sum\limits_{\textbf{e}}(\textbf{e}^{*}\textbf{D})(\textbf{e}\textbf{D})=(\bm{\nu}\times\textbf{D})(\bm{\nu}\times\textbf{D})$
(2)

the cross section in the resonant approximation can be written in the form:

$\displaystyle\sigma(\omega)=\frac{\mathrm{const}}{2J_{f}+1}\sum\limits_{J_{f}}\sum\limits_{M_{J_{i}}M_{J_{f}}}\left|\sum\limits_{\begin{subarray}{c}J_{n}M_{J_{n}}\\\
kJ_{k}M_{J_{k}}\end{subarray}}(E_{3^{3}D_{1}}-E_{2^{3}P_{J_{f}}})^{3/2}(E_{3^{3}D_{1}}-E_{2^{3}S_{1}}-\omega)^{1/2}\omega^{1/2}\right.$
(3) $\displaystyle\times\frac{\langle
2^{3}P_{J_{f}}|\bm{\nu}_{3}\times\textbf{D}|3^{3}D_{J_{n}}\rangle}{E_{3^{3}D_{J_{n}}}-E_{2^{3}S_{1}}-2\omega-\frac{\mathrm{i}}{2}\Gamma_{3^{3}D_{J_{n}}}}\left\\{\frac{\langle
3^{3}D_{J_{n}}|\textbf{e}_{2}\textbf{D}|k^{3}P_{J_{k}}\rangle\langle
k^{3}P_{J_{k}}|\textbf{e}_{1}\textbf{D}|2^{3}S_{1}\rangle}{E_{k^{3}P_{J_{k}}}-E_{2^{3}S_{1}}-\omega}\left.+\frac{\langle
3^{3}D_{J_{n}}|\textbf{e}_{1}\textbf{D}|k^{3}P_{J_{k}}\rangle\langle
k^{3}P_{J_{k}}|\textbf{e}_{2}\textbf{D}|2^{3}S_{1}\rangle}{E_{k^{3}P_{J_{k}}}-E_{3^{3}D_{J_{n}}}+\omega}\right\\}\right|^{2},$

where the regularization procedure established in Low (1952) and assembled
latter in Andreev  _et al._ (2008); Zalialiutdinov  _et al._ (2018) was
applied. This regularization procedure corresponds to the summation of an
infinite number of one-loop self-energy insertions into the electron
propagator. Within the framework of resonant approximation, this leads to the
appearance of gauge invariant level widths $\Gamma_{3^{3}D_{J_{n}}}$ in the
singular denominator Eq. (3), which, in turn, leads to the formation of an
absorption line profile.

To introduce NR corrections to the considered resonant transition $3^{3}D_{1}$
the remaining terms with $J_{n}=2$ and $J_{n}=3$ in the scattering amplitude
Eq. (3) should be considered. It can be found that the neighboring fine states
presented in the sum over $J_{n}$ provide the leading NR contribution
Jentschura and Mohr (2002), which is now known as the effect of quantum
interference. Then, following the methodology given in Anikin  _et al._
(2021), denoting the intervals $\Delta_{12}=E_{3^{3}D_{1}}-E_{3^{3}D_{2}}$ and
$\Delta_{13}=E_{3^{3}D_{1}}-E_{3^{3}D_{3}}$, summing over the projections of
final state and average over the projections of initial state, we arrive at

$\displaystyle\sigma(\omega)=\mathrm{const}\left[\frac{f_{\mathrm{res}}^{3^{3}D_{1}}}{(\omega_{0}-2\omega)^{2}+\frac{\Gamma_{3^{3}D_{1}}^{2}}{4}}\right.$
(4)
$\displaystyle+2\mathrm{Re}\frac{f_{\mathrm{nr}}^{3^{3}D_{1(2)}}}{(\omega_{0}-2\omega-\frac{\mathrm{i}\Gamma_{3^{3}D_{1}}}{2})(\omega_{0}+\Delta_{12}-2\omega)}$
$\displaystyle\left.+2\mathrm{Re}\frac{f_{\mathrm{nr}}^{3^{3}D_{1(3)}}}{(\omega_{0}-2\omega-\frac{\mathrm{i}\Gamma_{3^{3}D_{1}}}{2})(\omega_{0}+\Delta_{13}-2\omega)}\right],$

where we introduced following notations

$\displaystyle
f^{3^{3}D_{1}}_{\mathrm{res}}=\sum\limits_{M_{J_{i}}M_{J_{f}}}|T_{1}(\omega_{0}/2)|^{2},$
(5) $\displaystyle
f^{3^{3}D_{1(2)}}_{\mathrm{nr}}=\sum\limits_{M_{J_{i}}M_{J_{f}}}T_{1}(\omega_{0}/2)T^{*}_{2}(\omega_{0}/2),$
(6) $\displaystyle
f^{3^{3}D_{1(3)}}_{\mathrm{nr}}=\sum\limits_{M_{J_{i}}M_{J_{f}}}T_{1}(\omega_{0}/2)T^{*}_{3}(\omega_{0}/2),$
(7) $\displaystyle
T_{n}(\omega)=(E_{3^{3}D_{1}}-E_{2^{3}P_{J_{f}}})^{3/2}(E_{3^{3}D_{1}}-E_{2^{3}S_{1}}-\omega)^{1/2}\omega^{1/2}\sum\limits_{J_{n}M_{J_{n}}}\langle
2^{3}P_{J_{f}}|\textbf{e}_{3}^{*}\textbf{D}|3^{3}D_{J_{n}}\rangle$ (8)
$\displaystyle\times\sum\limits_{kJ_{k}M_{J_{k}}}\left\\{\frac{\langle
3^{3}D_{J_{n}}|\textbf{e}_{2}\textbf{D}|k^{3}P_{J_{k}}\rangle\langle
k^{3}P_{J_{k}}|\textbf{e}_{1}\textbf{D}|2^{3}S_{1}\rangle}{E_{k^{3}P_{J_{k}}}-E_{2^{3}S_{1}}-\omega}+\frac{\langle
3^{3}D_{J_{n}}|\textbf{e}_{1}\textbf{D}|k^{3}P_{J_{k}}\rangle\langle
k^{3}P_{J_{k}}|\textbf{e}_{2}\textbf{D}|2^{3}S_{1}\rangle}{E_{k^{3}P_{J_{k}}}-E_{3^{3}D_{J_{n}}}+\omega}\right\\}.$

Here $\omega_{0}=E_{3^{3}D_{1}}-E_{2^{3}S_{1}}$ is the two-photon transition
energy, $\Gamma_{3^{3}D_{1}}$ is the natural width of $3^{3}D_{1}$ state, and
$f_{\mathrm{res}}^{3^{3}D_{1}}$, $f_{\mathrm{nr}}^{3^{3}D_{1(2)}}$,
$f_{\mathrm{nr}}^{3^{3}D_{1(3)}}$ are coefficients that determine the
dependence on angles $\theta_{ij}$ ($i,j=1,\;2,\;3$), see Eqs. (13)-(17) and
Appendix A in Anikin  _et al._ (2021). According to Anikin  _et al._ (2021)
the coefficient $f_{\mathrm{res}}^{3^{3}D_{1}}$ describes the angular
correlation of leading resonant two-photon scattering process via intermediate
state with $J_{n}=1$, while $f_{\mathrm{nr}}^{3^{3}D_{1(2)}}$ (or
$f_{\mathrm{nr}}^{3^{3}D_{1(3)}}$ ) corresponds to the interference
contribution between fine sublevels with $J_{n}=1$ and $J_{n}=2$ (or $J_{n}=1$
and $J_{n}=3$).

The first Lorentzian term in Eq. (4) corresponds to the resonant transition,
while two latter terms describe the asymmetry of line profile and were not
considered in Dorrer  _et al._ (1997); Hlousek  _et al._ (1983). Consequently,
this leads to the shift of line position. Then the resonant transition
frequency $\omega_{\mathrm{res}}$ can be defined from $d\sigma(\omega)$ in one
of the evident ways as the maximum value of $d\sigma(\omega)$ using the
condition:

$\displaystyle\frac{d}{d\omega}\sigma(\omega)=0.$ (9)

In the resonant approximation, retaining only the first term in Eq. (4), we
find
$\omega_{\mathrm{res}}=\omega_{\mathrm{max}}=\omega_{0}/2=(E_{3^{3}D_{1}}-E_{2^{3}S_{1}})/2$.
Then, taking into account the two interference terms in Eq. (4) and solving
Eq. (9) with respect to $\omega$ one can find $\omega_{\mathrm{max}}$ as

$\displaystyle\omega_{\mathrm{max}}=(\omega_{0}-\delta_{\mathrm{NR}})/2,$ (10)

where

$\displaystyle\delta_{\mathrm{NR}}=\frac{f^{3^{3}D_{1(2)}}_{\mathrm{nr}}}{f^{3^{3}D_{1}}_{\mathrm{res}}}\frac{(\Gamma_{3^{3}D_{1}})^{2}}{4\Delta_{12}}+\frac{f^{3^{3}D_{1(3)}}_{\mathrm{nr}}}{f^{3^{3}D_{1}}_{\mathrm{res}}}\frac{(\Gamma_{3^{3}D_{1}})^{2}}{4\Delta_{13}}.$
(11)

The nonresonant correction Eq. (11) is obtained as the contribution of the
leading order in the series expansion in terms of the relation $\Gamma/\Delta$
Solovyev  _et al._ (2020a), when this parameter is small (in the considered
examples inequality $\Gamma/\Delta\ll 1$ always holds). Performing angular
algebra, we find

$\displaystyle
f^{3^{3}D_{1(2)}}_{\mathrm{nr}}/f^{3^{3}D_{1}}_{\mathrm{res}}=95/198,$ (12)

and

$\displaystyle
f^{3^{3}D_{1(3)}}_{\mathrm{nr}}/f^{3^{3}D_{1}}_{\mathrm{res}}=7/198.$ (13)

Equations (12) and (13) are calculated in a fully analytical way using
definitions Eqs. (5)-(8). The summation over the entire spectrum of
intermediate states is not required in the considered case since the
corresponding contribution vanishes in the ratio
$f_{\mathrm{nr}}/f_{\mathrm{res}}$. In the nonrelativistic limit sum over
index $k$ in the curly brackets of Eq. (3) (or Eq. (8)) becomes independent on
the values of total angular momenta $J_{k},\;J_{n},\;J_{f}$. Then after the
reduction of matrix elements with the use of Eckart-Wigner theorem the
corresponding reduced sums become equivalent for different values $J_{n}$ in
Eq. (8) and vanishes in the ratio $f_{\mathrm{nr}}/f_{\mathrm{res}}$.

Finally, substituting the values for natural line width
$\Gamma_{3^{3}D_{1}}=11.35(6)$ MHz Schmoranzer  _et al._ (1991) and energy
differences $\Delta_{12}=1325.025(33)$ MHz and $\Delta_{13}=1400.290(33)$ MHz
Yerokhin  _et al._ (2020) in Eq. (11), the NR correction to the transition
frequency $2^{3}S_{1}-3^{3}D_{1}$ is

$\displaystyle\delta_{\mathrm{NR}}=0.0124(4)\,\,\,{\rm MHz}.$ (14)

An analytical evaluation, Eqs. (12) and (13), shows that similarly to the NR
correction to the $2s_{1/2}^{F=1}\rightarrow nd_{3/2(5/2)}^{F=2}$ transition
frequencies in hydrogen Solovyev  _et al._ (2020b), the correction Eq. (11)
also does not depend on the angles between any pair of the vectors
$\bm{\nu}_{3}$, $\textbf{e}_{1}$ and $\textbf{e}_{2}$. Therefore, the
asymmetry of observed line profile cannot be eliminated by choosing ’magic’
angles or experimental geometry as in Beyer and et al. (2017); Yost  _et al._
(2014). An important outcome of these calculations, involving the natural
level width, is that the magnitude of effect is at the level of experimental
uncertainty $0.056$ MHz Dorrer  _et al._ (1997).

However, the actual experimental width $\Gamma^{\mathrm{exp}}$ of the observed
profile differs significantly from the natural one $\Gamma^{\mathrm{nat}}$ due
to different broadening mechanisms Riehle (2004). In fact, the width of level
in Eq. (4) must be associated with an experimental value de Beauvoir  _et al._
(2000); Labzowsky  _et al._ (2002). In Dorrer  _et al._ (1997), the main
broadening effects are due to pressure and transit time. Denoting two latter
contributions as $\Gamma^{\mathrm{pb}}$ and $\Gamma^{\mathrm{tt}}$,
respectively, the full width at half maximum in the experiment Dorrer  _et
al._ (1997) can be expressed as a sum of three contributions

$\displaystyle\Gamma^{\mathrm{exp}}=\Gamma^{\mathrm{nat}}+\Gamma^{\mathrm{pb}}+\Gamma^{\mathrm{tt}},$
(15)

where $\Gamma^{\mathrm{nat}}$ denotes the natural level width.

According to Dorrer  _et al._ (1997), the pressure broadening is parametrized
as $\Gamma^{\mathrm{pb}}/p=35.7(1.7)\;[\mathrm{MHz/Torr}]$, where $p$ is the
pressure in Torr. Typically, the absorption signal is measured at various
values of $p$ and then the result is extrapolated to zero pressure (in Dorrer
_et al._ (1997) the $p$ values are in the range of $0.05-0.5$ Torr).

The value of transit time broadening $\Gamma^{\mathrm{tt}}$ is not presented
in the experiment Dorrer  _et al._ (1997). However, we can roughly estimate
$\Gamma^{\mathrm{tt}}$ by attributing it to a difference between experimental
width extrapolated to zero pressure $\Gamma^{\mathrm{exp}}=11.33(19)$ MHz and
natural width $\Gamma^{\mathrm{nat}}=11.26$ MHz calculated theoretically Drake
(2006), that gives $\Gamma^{\mathrm{tt}}=0.07(19)$ MHz. Thus, one can conclude
that pressure broadening represents the main contribution to
$\Gamma^{\mathrm{exp}}$. Finally, for the pressures $p$ in the range from
$0.05$ to $0.5$ Torr the experimental width Eq. (15) of observed profile
belongs to the interval $\Gamma^{\mathrm{exp}}\in[13.2(4),29.2(4)]$ MHz.
Substitution of these values into Eq. (11) leads to the NR corrections
$\delta_{\mathrm{NR}}$ to transition frequency in the range from $0.016(1)$ to
$0.082(19)$ MHz. The latter value only partly removes the current discrepancy
between theoretical and experimental value of $2^{3}S_{1}-3^{3}D_{1}$
transition frequency which is of about $0.5$ MHz Patkóš  _et al._ (2021).

The observed fluorescent signal was also fitted at a pressure of $p=0.151$
Torr (see Fig. 1 in Dorrer  _et al._ (1997)). The NR correction corresponding
to $p=0.151$ Torr is $\delta_{\mathrm{NR}}=0.027$ MHz. This value still does
not remove the discrepancy between theoretical calculations and experiment
found in Patkóš  _et al._ (2021), however reaches the level of experimental
uncertainty $E^{\mathrm{exp}}(2^{3}S_{1}-3^{3}D_{1})=786\,823\,850.002(56)$
MHz Dorrer  _et al._ (1997). It is important to note that the resulting
experimental value of $2^{3}S_{1}-3^{3}D_{1}$ transition frequency in Dorrer
_et al._ (1997) was obtained by extrapolating the line position to zero
pressure, i.e. to the case when $\Gamma^{\mathrm{exp}}$ approximately equal
$\Gamma^{\mathrm{nat}}$. Then repairing the center of the line by NR
correction at each value of pressure in accordance with Eq. (10), the result
is expected to be different (see Fig. 2b in Dorrer  _et al._ (1997)),
eliminating the current discrepancy with theory, at least in part. We
schematically reproduce fit given in Dorrer  _et al._ (1997) and draw a new
fit accounting for NR frequency corrected to $\delta_{\mathrm{NR}}$ at each
value of pressure according to
$\omega^{\prime}_{0}=\omega_{0}-\delta_{\mathrm{NR}}$, see Fig. 1 (blue points
and corresponding line). The result of such extrapolation to zero pressure for
shifted value $\omega^{\prime}_{0}$ is, however, on the level of experimental
uncertainty. Subtracting $\delta_{\mathrm{NR}}$ from experimental points
$\omega_{0}$ in Fig. 1 (blue points) we proceed from the fact that in
experiment Dorrer  _et al._ (1997) the nonresonant corrections were not
considered, which means that they are included in the measured value (red
points in Fig. 1).

Figure 1: Extrapolations of line position for $2^{3}S_{1}-3^{3}D_{1}$
transition versus the pressure. The blue points take into account the
corresponding NR corrections to the original data (red dots), given in
accordance with Eq. (10). The corresponding dashed blue line represents the
extrapolation to zero pressure through the points that take into account the
NR effects. Likewise, the solid red line extrapolates uncorrected points.
Uncertainties are also taken from Dorrer  _et al._ (1997). The frequency
points take into account the second order Doppler effect, which turned out to
be 8.1 KHz at T=300 K Dorrer  _et al._ (1997). The least squares approach is
used for extrapolation.

Similar calculations of NR corrections can be done for the measurement of
$2^{3}S_{1}-4^{3}D_{1}$ and $2^{3}S_{1}-5^{3}D_{1}$ transition frequencies
presented in Hlousek  _et al._ (1983). These experiments were carried out
using the same technique as Dorrer  _et al._ (1997) resulting to the values
$\Delta E^{\mathrm{exp}}(2^{3}S_{1}-4^{3}D_{1})=947\,000\,197.11(1.8)$ MHz and
$\Delta E^{\mathrm{exp}}(2^{3}S_{1}-5^{3}D_{1})=102\,112\,869\,7.31(2.4)$ MHz.
The pressure-broadening coefficients of $2^{3}S_{1}-4^{3}D_{1}$ and
$2^{3}S_{1}-5^{3}D_{1}$ lines were determined as
$\Gamma^{\mathrm{pb}}/p=68.1(2.7)[\mathrm{MHz/Torr}]$ and
$\Gamma^{\mathrm{pb}}/p=78.5(2.7)[\mathrm{MHz/Torr}]$, respectively (see Table
III in Hlousek  _et al._ (1983)), while transit-time effects are supposed to
be negligible. Then using the values of the corresponding fine structure
intervals $\Delta_{12}=E_{4^{3}D_{1}}-E_{4^{3}D_{2}}=555.231(7)$ MHz,
$\Delta_{13}=E_{4^{3}D_{1}}-E_{4^{3}D_{3}}=591.253(6)$ MHz Morton  _et al._
(2006), and the natural level width
$\Gamma_{4^{3}D_{1}}^{\mathrm{nat}}=4.96274$ MHz Theodosiou (1984) we find the
NR correction to the transition frequency $2^{3}S_{1}-4^{3}D_{1}$ in the range
from $0.350(23)$ to $2.65(2)$ MHz for pressure $p=0.5-1.5$ Torr. The indicated
corrections are at the level or exceed the experimental uncertainty for
$2^{3}S_{1}-4^{3}D_{1}$ transition frequency which is $1.8$ MHz. Likewise, the
NR correction values for the $2^{3}S_{1}-5^{3}D_{1}$ transition frequency
start at $0.79(5)$ and end at $6.6(4)$ MHz for the energy intervals
$\Delta_{12}=E_{5^{3}D_{1}}-E_{5^{3}D_{2}}=283.560(8)$ MHz,
$\Delta_{13}=E_{5^{3}D_{1}}-E_{5^{3}D_{3}}=302.781(8)$ MHz Morton  _et al._
(2006), and the natural level width
$\Gamma_{5^{3}D_{1}}^{\mathrm{nat}}=2.61381$ MHz Theodosiou (1984). The
results for all considered examples are summarized in Table 1. It is should be
noted that considered NR corrections to $2^{3}S_{1}-n^{3}D_{1}$ ($n=4,\;5$)
transition frequencies are higher than for $2^{3}S_{1}-3^{3}D_{1}$ transition
frequency due to the smaller value of the corresponding fine structure
intervals.

Table 1: Range of nonresonant corrections to $2^{3}S_{1}-n^{3}D_{1}$ ($n=3,\;4,\;5$) (5th column) transition frequencies (2nd column) at different ranges of experimental transition width (4th column), see Eq. (15). All values are in MHz. Uncertainties are given in brackets. Transition | Experiment, MHz Dorrer  _et al._ (1997); Hlousek  _et al._ (1983) | Theory, MHz Yerokhin  _et al._ (2020); Patkóš  _et al._ (2021) | Experimental widths $\Gamma^{\mathrm{exp}}$, MHz | NR corrections $\delta_{\mathrm{NR}}$, MHz
---|---|---|---|---
$2^{3}S_{1}-3^{3}D_{1}$ | $786\,823\,850.002(56)$ | $786\,823\,849.540(57)$ | $13.2(4)-29.2(4)$ | $0.016(1)-0.082(19)$
$2^{3}S_{1}-4^{3}D_{1}$ | $947\,000\,197.11(1.8)$ | $947\,000\,194.44(5)$ | $39(1.4)-107(4)$ | $0.350(23)-2.65(2)$
$2^{3}S_{1}-5^{3}D_{1}$ | $102\,112\,869\,7.31(2.4)$ | $102\,112\,869\,8.36(5)$ | $41.9(1.4)-120(4)$ | $0.79(5)-6.6(4)$

Summing up the results of this work, it can be argued that the achieved
measurement accuracy of the transition frequencies in helium has reached a
level at which the nonresonant corrections and, in particular, the effect of
quantum interference as their most significant part could be observed in two-
photon atomic spectroscopy of helium. Since the NR correction found here is
dependent on experimental conditions, each such calculation must take into
account the corresponding features of experiment. The above analysis shows
that the experimentally observed line broadening can play a key role in
determining the nonresonant correction and, as a consequence, in determining
the value of transition frequency. A decrease in the experimentally observed
linewidth to the natural one should significantly reduce the risk of detecting
discrepancies between the experimental and theoretical values of the
transition energies.

Despite the fact that the experiments Dorrer  _et al._ (1997); Hlousek  _et
al._ (1983) do not take into account the nonresonant effects, taking them into
account still cannot completely remove the currently existing discrepancies
with the theoretical predictions of the transition frequencies. Then it can be
expected that the reconciliation between experiment and theory is most likely
related to the issue of more accurate measurement or precise QED calculations
of unaccounted contributions.

##  I Acknowledgements

This work was supported by Russian Science Foundation (Grant No. 20-72-00003).

## References

  * Beyer and et al. (2017) A. Beyer and et al., Science 358, 79 (2017).
  * Yost  _et al._ (2014) D. C. Yost, A. Matveev, E. Peters, A. Beyer, T. W. Hänsch, and T. Udem, Phys. Rev. A 90, 012512 (2014).
  * Amaro  _et al._ (2015) P. Amaro, B. Franke, J. J. Krauth, M. Diepold, F. Fratini, L. Safari, J. Machado, A. Antognini, F. Kottmann, P. Indelicato, R. Pohl, and J. P. Santos, Phys. Rev. A 92, 022514 (2015).
  * Brown  _et al._ (2013) R. C. Brown, S. Wu, J. V. Porto, C. J. Sansonetti, C. E. Simien, S. M. Brewer, J. N. Tan, and J. D. Gillaspy, Phys. Rev. A 87, 032504 (2013).
  * Marsman  _et al._ (2015) A. Marsman, M. Horbatsch, and E. A. Hessels, J. Phys. Chem. Ref. Data 44, 031207 (2015).
  * Sansonetti  _et al._ (2011) C. J. Sansonetti, C. E. Simien, J. D. Gillaspy, J. N. Tan, S. M. Brewer, R. C. Brown, S. Wu, and J. V. Porto, Phys. Rev. Lett. 107, 023001 (2011).
  * Jentschura and Mohr (2002) U. D. Jentschura and P. J. Mohr, Can. J. of Phys. 80, 633 (2002).
  * Patkóš  _et al._ (2021) V. Patkóš, V. A. Yerokhin, and K. Pachucki, Phys. Rev. A 103, 042809 (2021).
  * Yerokhin  _et al._ (2020) V. A. Yerokhin, V. Patkóš, M. Puchalski, and K. Pachucki, Phys. Rev. A 102, 012807 (2020).
  * Dorrer  _et al._ (1997) C. Dorrer, F. Nez, B. de Beauvoir, L. Julien, and F. Biraben, Phys. Rev. Lett. 78, 3658 (1997).
  * Schwob  _et al._ (1999) C. Schwob, L. Jozefowski, O. Acef, L. Hilico, B. de Beauvoir, F. Nez, L. Julien, A. Clairon, and F. Biraben, IEEE Transactions on Instrumentation and Measurement 48, 178 (1999).
  * Labzowsky  _et al._ (2001) L. Labzowsky, D. A. Solovyev, G. Plunien, and G. Soff, Phys. Rev. Lett. 87, 143003 (2001).
  * Anikin  _et al._ (2021) A. Anikin, T. Zalialiutdinov, and D. Solovyev, Phys. Rev. A 103, 022833 (2021).
  * Andreev  _et al._ (2008) O. Y. Andreev, L. N. Labzowsky, G. Plunien, and D. A. Solovyev, Phys. Rep. 455, 135 (2008).
  * Drake  _et al._ (1969) G. W. F. Drake, G. A. Victor, and A. Dalgarno, Phys. Rev. 180, 25 (1969).
  * Zalialiutdinov  _et al._ (2016) T. Zalialiutdinov, D. Solovyev, L. Labzowsky, and G. Plunien, Phys. Rev. A 93, 012510 (2016).
  * Labzowsky  _et al._ (1993) L. Labzowsky, G. Klimchitskaya, and Y. Dmitriev,  _Relativistic Effects in the Spectra of Atomic Systems_ (Institute of Physics Publishing, 1993).
  * Solovyev  _et al._ (2020a) D. Solovyev, A. Anikin, T. Zalialiutdinov, and L. Labzowsky, Journal of Physics B: Atomic, Molecular and Optical Physics 53, 125002 (2020a).
  * Low (1952) F. Low, Phys. Rev. 88, 53 (1952).
  * Zalialiutdinov  _et al._ (2018) T. A. Zalialiutdinov, D. A. Solovyev, L. N. Labzowsky, and G. Plunien, Phys. Rep. 737, 1 (2018).
  * Hlousek  _et al._ (1983) L. Hlousek, S. A. Lee, and W. M. Fairbank, Phys. Rev. Lett. 50, 328 (1983).
  * Schmoranzer  _et al._ (1991) H. Schmoranzer, H. Roth, U. Volz, and D. Marger, Journal of Physics B: Atomic, Molecular and Optical Physics 24, 595 (1991).
  * Solovyev  _et al._ (2020b) D. Solovyev, T. Zalialiutdinov, and A. Anikin, “Proton size from precision experiments on hydrogen and muonic hydrogen atoms,” (2020b), arXiv:2009.11520 [physics.atom-ph] .
  * Riehle (2004) F. Riehle,  _Frequency Standards: Basics and Applications_ (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2004).
  * de Beauvoir  _et al._ (2000) B. de Beauvoir, C. Schwob, O. Acef, L. Jozefowski, L. Hilico, F. Nez, L. Julien, A. Clairon, and F. Biraben, Eur. Phys. J. D 12, 61 (2000).
  * Labzowsky  _et al._ (2002) L. Labzowsky, D. Soloviev, G. Plunien, and G. Soff, Phys. Rev. A 65, 054502 (2002).
  * Drake (2006) G. W. F. Drake,  _Springer Handbook of Atomic, Molecular, and Optical Physics_, Springer Handbook of Atomic, Molecular, and Optical Physics, Edited by Gordon W.F. Drake. 2006 LVIII, 1504 p. 288 illus. in color, With CD-ROM. 2nd ed. 0-387-20802-X. Berlin: Springer, 2006. (Springer, 2006).
  * Morton  _et al._ (2006) D. Morton, Q. Wu, and G. Drake, Canadian Journal of Physics 84 (2006), 10.1139/p06-009.
  * Theodosiou (1984) C. E. Theodosiou, Phys. Rev. A 30, 2910 (1984).


# Commutativity and spectral properties for a general class of Szász-Mirakjan-
Durrmeyer operators

Ulrich Abel  Ana Maria Acu  Margareta Heilmann  Ioan Raşa  Fachbereich MND
Technische Hochschule Mittelhessen, Germany, e-mail<EMAIL_ADDRESS>Department of Mathematics and Informatics, Lucian Blaga University of Sibiu,
Romania (Corresponding Author), e-mail<EMAIL_ADDRESS>School of
Mathematics and Natural Sciences, University of Wuppertal, Germany, e-mail:
<EMAIL_ADDRESS>Technical University of Cluj-Napoca, Faculty
of Automation and Computer Science, Department of Mathematics, Str.
Memorandumului nr. 28, 400114 Cluj-Napoca, Romania e-mail:
<EMAIL_ADDRESS>
###### Abstract

In this paper we present commutativity results for a general class of Szász-
Mirakjan-Durrmeyer type operators and associated differential operators and
investigate their eigenfunctions.

###### keywords:

Positive linear operators, Szász-Mirakjan-Durrmeyer type operators,
commutativity, differential operators, spectral properties

###### MSC:

[2020] 41A36.

main.tex

##  1 Introduction

For fixed $j\in\mathbb{Z}$ we consider sequences of positive linear operators
$S_{n,j}$ which can be viewed as a generalization of the Szász-Mirakjan-
Durrmeyer operators [11], Phillips operators [12] and corresponding
Kantorovich modifications of higher order.

For $j\in{\mathbb{N}}$, these operators possess the exceptional property to
preserve constants and the monomial $x^{j}$.

It turned out (see [1]), that an extension of this family covers certain well-
known operators studied before, so that the outcoming results could be
unified. Continuous functions of exponential growth on the positive half-axis
are approximated by the operators $S_{n,j}$, $j\in\mathbb{Z}$ in each point of
continuity (see [1, Corollary 12].

In this paper we investigate commutativity properties of the operators and
their commutativity with certain differential operators as well as their
spectral properties.

For $A\geq 0$, we denote by $E_{A}$ the space of functions
$f:[0,\infty)\longrightarrow\mathbb{R}$, $f$ locally integrable, satisfying
the growth condition $\left|f\left(t\right)\right|\leq Ke^{At}$, $t\geq 0$ for
some positive constant $K$. Furthermore, define
$E:=\displaystyle\bigcup_{A\geq 0}E_{A}$.

Throughout the paper we use the convention that a sum is zero if the upper
limit is smaller than the lower one.

###### Definition 1

Let $f\in E_{A}$, $n>A$ and $j\in{\mathbb{Z}}$. Then the operators $S_{n,j}$
are defined by

$(S_{n,j}f)(x)=f(0)\sum_{k=0}^{j-1}s_{n,k}(x)+\sum_{k=j}^{\infty}s_{n,k}(x)n\int_{0}^{\infty}s_{n,k-j}(t)f(t)dt$

where

$s_{n,k}(x)=\dfrac{(nx)^{k}}{k!}e^{-nx},\,k\geq 0.$

For the sake of simplicity we define $s_{n,k}(x)=0$, if $k<0$.

For $j\leq 0$ we have

$(S_{n,j}f)(x)=\sum_{k=0}^{\infty}s_{n,k}(x)n\int_{0}^{\infty}s_{n,k-j}(t)f(t)dt.$

For the special case $j=0$ we get the Szász-Mirakjan-Durrmeyer operators first
defined in [11], for $j=1$ the Phillips operators, also called genuine Szász-
Mirakjan-Durrmeyer operators, [12]. For $j\leq-1$ the operators coincide with
certain auxiliary operators, see, e.g., [8, (1.4)] for $c=0$, named $M_{n,r}$
there, $r=-j$ or [6, (3.5)] for $c=0$, named $M_{n,s}$ there, $s=-j$. They can
also be considered as corresponding Kantorovich modifications of higher order.

In [1, Lemma 2] it was proved that $S_{n,j}f\in E_{2A}$ for $n>2A$, $A\geq 0$,
$f\in E_{A}$.

For $l=0,1,\ldots$ we denote $e_{l}(t)=t^{l}$, $t\geq 0$.

All these operators preserve $e_{0}$ and for $j\geq 1$ also $e_{j}$ since

$\displaystyle(S_{n,j}e_{j})(x)$ $\displaystyle=$
$\displaystyle\sum_{k=j}^{\infty}s_{n,k}(x){n\int_{0}^{\infty}s_{n,k-j}(t)t^{j}dt}$
$\displaystyle=$ $\displaystyle x^{j}\sum_{k=j}^{\infty}s_{n,k-j}(x)=x^{j}.$

Furthermore, for $j\geq 1$ it is easy to see that $S_{n,j}(f;0)=f(0)$.

Throughout the paper we use the following formulas

$\displaystyle s_{n,k}^{\prime}(x)$ $\displaystyle=$ $\displaystyle
n(s_{n,k-1}(x)-s_{n,k}(x)),$ (1)
$\displaystyle\int_{0}^{\infty}s_{n,l}(t)t^{r}dt$ $\displaystyle=$
$\displaystyle\frac{(l+r)!}{l!}n^{-r-1},$ (2)
$\displaystyle\int_{0}^{\infty}s_{m,r}(t)s_{n,l}(t)dt$ $\displaystyle=$
$\displaystyle\frac{m^{r}n^{l}}{(m+n)^{r+l+1}}\frac{(r+l)!}{r!l!}.$ (3)

##  2 Commutativity of the operators

In this section our aim is to show that the operators commute for a fixed $j$.
For $j=0$ this was proved in [4]. Indeed, here we prove a more general result
for compositions of the operators from which the commutativity then follows as
a corollary as well as a nice representation for iterates of $S_{n,j}$. For
$j=0$ the following result was proved in [2], for $j=1$ see [9, Theorem 3.1]
and for $j\leq 0$ see [13, Theorem 1].

Due to [1, Remark 3] we obtain

$\left|S_{m,j}\left(S_{n,j}f\right)\left(x\right)\right|\leq
1+\left(\frac{n-A}{n}\right)^{j-1}\left(\frac{m-\frac{An}{n-A}}{m}\right)^{j-1}\exp\left(\frac{Am}{m-\frac{An}{n-A}}x\right)$

for each integer $j$, $n>A$ and $m>\frac{An}{n-A}$.

For $n>A$ the inequality $m>\frac{An}{n-A}$ is equivalent to
$\frac{mn}{m+n}>A$.

###### Theorem 2

Let $f\in E_{A}$, $n>A$ and $m>\frac{An}{n-A}$. Then

$S_{m,j}(S_{n,j}f)=S_{\frac{mn}{m+n},j}f.$ (4)

Proof. Due to the preceding remarks we only have to prove the theorem for
$j\geq 2$. The considerations above show that both sides of (4) are well
defined. According to [1, Lemma 2], $S_{m,j}$ can be applied to $S_{n,j}f$.

Now we can write

$\displaystyle(S_{\frac{mn}{m+n},j}f)(x)$ $\displaystyle=$ $\displaystyle
f(0)\sum_{k=0}^{j-1}s_{\frac{mn}{m+n},k}(x)+\sum_{k=j}^{\infty}s_{\frac{mn}{m+n},k}(x)\cdot\frac{nm}{n+m}\cdot\int_{0}^{\infty}s_{\frac{mn}{m+n},k-j}(t)f(t)dt$
$\displaystyle=:$ $\displaystyle T_{1}+T_{2}$ $\displaystyle(S_{n,j}f)(x)$
$\displaystyle=$ $\displaystyle
f(0)\sum_{k=0}^{j-1}s_{n,k}(x)+\sum_{k=j}^{\infty}s_{n,k}(x)\cdot
n\cdot\int_{0}^{\infty}s_{n,k-j}(t)f(t)dt$ $\displaystyle=:$
$\displaystyle(S_{n,j,1}f)(x)+(S_{n,j,2}f)(x)$
$\displaystyle(S_{m,j}(S_{n,j}f)(y))(x)$ $\displaystyle=$
$\displaystyle(S_{m,j,1}(S_{n,j,1}f)(y))(x)+(S_{m,j,2}(S_{n,j,1}f)(y))(x)$
$\displaystyle+(S_{m,j,1}(S_{n,j,2}f)(y))(x)+(S_{m,j,2}(S_{n,j,2}f)(y))(x)$
$\displaystyle=:$ $\displaystyle T_{3}+T_{4}+T_{5}+T_{6}.$

It is easy to verify that $T_{5}=0$. We will now prove that $T_{6}=T_{2}$ and
$T_{3}+T_{4}=T_{1}$.

With (3) we get

$\displaystyle T_{6}$ $\displaystyle=$
$\displaystyle\sum_{k=j}^{\infty}s_{m,k}(x)m\int_{0}^{\infty}s_{m,k-j}(t)\left[\sum_{l=j}^{\infty}s_{n,l}(t)n\int_{0}^{\infty}s_{n,l-j}(y)f(y)dy\right]dt$
(5) $\displaystyle=$ $\displaystyle
mn\int_{0}^{\infty}\left\\{f(y)\sum_{k=j}^{\infty}\sum_{l=j}^{\infty}s_{m,k}(x)s_{n,l-j}(y){\int_{0}^{\infty}s_{m,k-j}(t)s_{n,l}(t)dt}\right\\}dy$
$\displaystyle=:$ $\displaystyle\int_{0}^{\infty}f(y)T_{m,n}(x,y)dy.$
$\displaystyle T_{m,n}(x,y)$ $\displaystyle=$ $\displaystyle
mn\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}s_{m,k+j}(x)s_{n,l}(y)\frac{m^{k}n^{l+j}}{(m+n)^{k+l+1+j}}\cdot\frac{(k+l+j)!}{k!(l+j)!}$
$\displaystyle=$
$\displaystyle\frac{mn}{m+n}e^{-mx}e^{-ny}\sum_{k=0}^{\infty}\frac{m^{2k+j}x^{k+j}}{(m+n)^{k+j}}\frac{1}{k!(k+j)!}T(y),$

where

$\displaystyle T(y)$ $\displaystyle:=$
$\displaystyle\sum_{l=0}^{\infty}\frac{n^{2l+j}y^{l}}{(m+n)^{l}}\frac{(k+l+j)!}{l!(l+j)!}$
$\displaystyle=$ $\displaystyle
n^{j}\sum_{l=0}^{\infty}\frac{(\frac{n^{2}}{m+n}y)^{l}}{l!(l+j)!}\cdot(k+l+j)!.$

Using the relation

$\displaystyle\sum_{l=0}^{\infty}\frac{(l+k+j)!}{(l+j)!}\frac{z^{l}}{l!}$
$\displaystyle=$ $\displaystyle
z^{-j}\left(\frac{d}{dz}\right)^{k}\sum_{l=0}^{\infty}\frac{z^{l+k+j}}{l!}$
$\displaystyle=$ $\displaystyle
z^{-j}\left(\frac{d}{dz}\right)^{k}\left(z^{k+j}e^{z}\right)$ $\displaystyle=$
$\displaystyle
e^{z}\sum_{\nu=0}^{k}{\binom{k}{\nu}}\frac{(k+j)!}{(\nu+j)!}z^{\nu}$

with $z=\frac{n^{2}}{m+n}y$ we obtain

$T(y)=n^{j}e^{\frac{n^{2}}{m+n}y}\sum_{\nu=0}^{k}{\binom{k}{\nu}}\frac{(k+j)!}{(\nu+j)!}\left(\frac{n^{2}}{m+n}y\right)^{\nu}.$

Inserting this in $T_{m,n}(x,y)$ above we get

$\displaystyle T_{m,n}(x,y)$
$\displaystyle=\frac{n^{j+1}m}{m+n}e^{-mx}e^{-ny}e^{\frac{n^{2}}{m+n}y}\sum_{k=0}^{\infty}\frac{m^{2k+j}x^{k+j}}{(m+n)^{k+j}}\frac{1}{k!(k+j)!}\sum_{\nu=0}^{k}{\binom{k}{\nu}}\frac{(k+j)!}{(\nu+j)!}\left(\frac{n^{2}}{m+n}y\right)^{\nu}$
$\displaystyle=\frac{n^{j+1}m}{m+n}e^{-mx}e^{-ny}e^{\frac{n^{2}}{m+n}y}\sum_{\nu=0}^{\infty}\frac{1}{\nu!(\nu+j)!}\left(\frac{n^{2}}{m+n}y\right)^{\nu}\sum_{k=\nu}^{\infty}\frac{1}{(k-\nu)!}\frac{m^{2k+j}x^{k+j}}{(m+n)^{k+j}}$

Now we calculate

$\displaystyle\sum_{k=\nu}^{\infty}\frac{1}{(k-\nu)!}\frac{m^{2k+j}x^{k+j}}{(m+n)^{k+j}}$
$\displaystyle=$
$\displaystyle\frac{m^{2\nu+j}x^{\nu+j}}{(m+n)^{\nu+j}}\sum_{k=0}^{\infty}\frac{\left(\frac{m^{2}x}{m+n}\right)^{k}}{k!}$
$\displaystyle=$
$\displaystyle\frac{m^{2\nu+j}x^{\nu+j}}{(m+n)^{\nu+j}}e^{\frac{m^{2}}{m+n}x}$

which gives

$\displaystyle T_{m,n}(x,y)$ $\displaystyle=$
$\displaystyle\frac{nm}{m+n}e^{-\frac{mn}{m+n}x}e^{-\frac{mn}{m+n}y}\sum_{\nu=0}^{\infty}\frac{\left(\frac{mn}{m+n}x\right)^{\nu+j}}{(\nu+j)!}\frac{\left(\frac{mn}{m+n}y\right)^{\nu}}{\nu!}$
$\displaystyle=$
$\displaystyle\frac{mn}{m+n}\sum_{\nu=0}^{\infty}s_{\frac{mn}{n+m},\nu+j}(x)s_{\frac{mn}{n+m},\nu}(y).$

Thus with (5)

$T_{6}=T_{2}$

and it remains to prove that

$T_{3}+T_{4}=T_{1}.$

We have

$\displaystyle T_{3}$ $\displaystyle=$ $\displaystyle
f(0)\sum_{l=0}^{j-1}s_{n,l}(0)\sum_{k=0}^{j-1}s_{m,k}(x)$ $\displaystyle=$
$\displaystyle f(0)\sum_{k=0}^{j-1}s_{m,k}(x)$

and by using (3)

$\displaystyle T_{4}$ $\displaystyle=$
$\displaystyle\sum_{k=j}^{\infty}s_{m,k}(x)\cdot
m\cdot\int_{0}^{\infty}s_{m,k-j}(t)\left[f(0)\sum_{l=0}^{j-1}s_{n,l}(t)\right]dt$
$\displaystyle=$ $\displaystyle
f(0)m\sum_{k=j}^{\infty}s_{m,k}(x)\sum_{l=0}^{j-1}\int_{0}^{\infty}s_{m,k-j}(t)s_{n,l}(t)dt$
$\displaystyle=$ $\displaystyle
f(0)m\sum_{k=j}^{\infty}s_{m,k}(x)\sum_{l=0}^{j-1}\frac{(k+l-j)!}{(k-j)!l!}\cdot\frac{m^{k-j}n^{l}}{(m+n)^{k+l-j+1}}$
$\displaystyle=$ $\displaystyle
f(0)\frac{m}{m+n}e^{-mx}\sum_{l=0}^{j-1}\frac{1}{l!}\left(\frac{n}{m+n}\right)^{l}\left(\frac{m}{m+n}\right)^{-j}\sum_{k=j}^{\infty}\frac{(mx)^{k}}{k!}\frac{(k+l-j)!}{(k-j)!}\cdot\left(\frac{m}{m+n}\right)^{k}.$

With Leibniz formula for differentiation we can write

$\displaystyle\sum_{k=j}^{\infty}\frac{z^{k}}{k!}\frac{(k+l-j)!}{(k-j)!}$
$\displaystyle=$ $\displaystyle
z^{j}\sum_{k=j}^{\infty}\frac{1}{k!}\left(z^{k+l-j}\right)^{(l)}$
$\displaystyle=$ $\displaystyle
z^{j}\frac{d^{l}}{dz^{l}}\left\\{z^{l-j}\left[e^{z}-\sum_{k=0}^{j-1}\frac{z^{k}}{k!}\right]\right\\}$
$\displaystyle=$ $\displaystyle
z^{j}\sum_{i=0}^{l}{\binom{l}{i}}(l-i)^{\underline{l-i}}z^{i-j}\left[e^{z}-\sum_{k=i}^{j-1}\frac{z^{k-i}}{(k-i)!}\right]$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{l}{\binom{l}{i}}(-1)^{l-i}(j-i-1)^{\underline{l-i}}z^{i}\left[e^{z}-\sum_{k=i}^{j-1}\frac{z^{k-i}}{(k-i)!}\right].$

From this equation by setting $z=\frac{m^{2}x}{m+n}$ we get after
interchanging the order of summation

$\displaystyle T_{4}$
$\displaystyle=f(0)\left(\frac{m}{m+n}\right)^{1-j}e^{-mx}\sum_{i=0}^{j-1}\frac{1}{i!}\left(\frac{m^{2}x}{m+n}\right)^{i}\sum_{l=i}^{j-1}{\binom{j-i-1}{l-i}}(-1)^{l-i}\left(\frac{n}{m+n}\right)^{l}$
$\displaystyle\qquad\qquad\qquad\times\left\\{e^{\frac{m^{2}x}{m+n}}-\sum_{k=i}^{j-1}\frac{\left(\frac{m^{2}x}{m+n}\right)^{k-i}}{(k-i)!}\right\\}.$

With

$\sum_{l=i}^{j-1}{\binom{j-i-1}{l-i}}(-1)^{l-i}\left(\frac{n}{m+n}\right)^{l}=\left(\frac{n}{m+n}\right)^{i}\left(\frac{m}{m+n}\right)^{j-i-1}$

we derive

$\displaystyle T_{4}$ $\displaystyle=$ $\displaystyle
f(0)e^{-mx}\sum_{i=0}^{j-1}\frac{\left(\frac{m^{2}x}{m+n}\right)^{i}}{i!}\left(\frac{n}{m}\right)^{i}\left\\{e^{\frac{m^{2}x}{m+n}}-\sum_{k=i}^{j-1}\frac{\left(\frac{m^{2}x}{m+n}\right)^{k-i}}{(k-i)!}\right\\}$
$\displaystyle=$ $\displaystyle
f(0)\left\\{\sum_{i=0}^{j-1}s_{\frac{nm}{n+m},i}(x)-e^{-mx}\sum_{k=0}^{j-1}\frac{\left(\frac{m^{2}x}{m+n}\right)^{k}}{k!}\sum_{i=0}^{k}{\binom{k}{i}}\left(\frac{n}{m}\right)^{i}\right\\}$
$\displaystyle=$ $\displaystyle
f(0)\left\\{\sum_{i=0}^{j-1}s_{\frac{nm}{n+m},i}(x)-\sum_{k=0}^{j-1}s_{m,k}(x)\right\\}$
$\displaystyle=$ $\displaystyle T_{1}-T_{3}.$

Now we get the desired commutativity of the operators and the representation
for their iterates as a corollary.

###### Corollary 3

For all $f\in E_{A}$, $n,m>2A$ we have

$S_{m,j}({S}_{n,j}f)={S}_{n,j}({S}_{m,j}f)$

and for $l\in\mathbb{N}$, $m>lA$,

$S_{m,j}^{l}f=S_{\frac{m}{l},j}f.$ (6)

###### Remark 4

From (6) it is easy to see that

$I-\left(I-S_{n,j}\right)^{m+1}=\sum_{i=0}^{m}(-1)^{i}{\binom{m+1}{i+1}}S_{n,j}^{i+1}=\sum_{i=0}^{m}(-1)^{i}{\binom{m+1}{i+1}}S_{\frac{n}{i+1},j},$

i.e., iterative combinations are linear combinations. For $j=1$ see [10].

##  3 Derivatives and differential operators

As already known for $j=0$ and $j=1$ the operators $S_{n,j}$ are strongly
connected to appropriate differential operators which can be used to state
asymptotic relations in a concise form. In Theorem 8 we will show that
$S_{n,j}$ commutes with the corresponding differential operator, already
defined in [1, (3)] for sufficiently smooth functions. This property was first
proved for $j=0$ in [6, Lemma 3.1] and for $j=1$ in [9, Theorem 3.2]. For
$j\leq 1$ see also [7, Section 5] and the references therein.

For the proof of Theorem 8 we need the following representation for the
derivatives of $S_{n,j}f$ which can be found for $j=0$ in [5, (3.11)] or [6,
(3.1)]. For $j\leq 0$ see [13, Lemma 4].

###### Proposition 5

Let $l\in{\mathbb{N}}$ and $f^{(l)}\in E_{A}$, $A>0$ and $n>A$. Then

$\displaystyle(S_{n,j}f)^{(l)}(x)$ $\displaystyle=$ $\displaystyle
n\sum_{\nu=1}^{\min{\\{j-1,l-1\\}}}s_{n,j-1-\nu}^{(l-1-\nu)}(x)f^{(\nu)}(0)$
$\displaystyle+n\sum_{k=\max{\\{0,j-l\\}}}^{\infty}s_{n,k}(x)\int_{0}^{\infty}s_{n,k+l-j}(t)f^{(l)}(t)dt.$

###### Remark 6

It is easy to see that $f^{(l)}\in E_{A}$, $A>0$ implies $f\in E_{A}$. In the
case $A=0$ we have $f\in E_{\tilde{A}}$ for arbitrary $\tilde{A}>0$.

###### Remark 7

For $j\leq 1$ we get the identity

$(S_{n,j}f)^{(l)}=S_{n,j-l}f^{(l)}.$

Proof. Note that $f^{(l)}\in E_{A}$ implies $f\in E_{A}$.

We proceed by induction on $l$.
$l=1$: Using (1), index transform and again (1) we get

$\displaystyle(S_{n,j}f)^{\prime}(x)$ $\displaystyle=$ $\displaystyle
nf(0)s_{n,j-1}(x)-ns_{n,j-1}(x)\int_{0}^{\infty}s_{n,0}^{\prime}(t)f(t)dt$
$\displaystyle-n\sum_{k=j}^{\infty}s_{n,k}(x)\int_{0}^{\infty}s_{n,k+1-j}^{\prime}(t)f(t)dt$
$\displaystyle=$ $\displaystyle
n\sum_{k=j-1}^{\infty}s_{n,k}(x)\int_{0}^{\infty}s_{n,k+1-j}(t)f^{\prime}(t)dt\,,$

with integration by parts for the last equality.
$l\Rightarrow l+1$

$\displaystyle(S_{n,j}f)^{(l+1)}(x)$ $\displaystyle=$ $\displaystyle
n\sum_{\nu=1}^{\min{\\{j-1,l-1\\}}}s_{n,j-1-\nu}^{(l-\nu)}(x)f^{(\nu)}(0)$
$\displaystyle+n\sum_{k=\max{\\{0,j-l\\}}}^{\infty}s_{n,k}^{\prime}(x)\int_{0}^{\infty}s_{n,k+l-j}(t)f^{(l)}(t)dt$
$\displaystyle=:$ $\displaystyle S_{1}+S_{2}$

Using (1), index transform and again (1) we get

$\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle-n\sum_{k=\max{\\{0,j-(l+1)\\}}}^{\infty}s_{n,k}(x)\int_{0}^{\infty}s_{n,k+l+1-j}^{\prime}f^{(l)}(t)dt$
$\displaystyle=$ $\displaystyle
n\sum_{k=\max{\\{0,j-(l+1)\\}}}^{\infty}s_{n,k}(x)\int_{0}^{\infty}s_{n,k+l+1-j}(t)f^{(l+1)}(t)dt$
$\displaystyle+ns_{n,j-(l+1)}(x)f^{(l)}(0),$

again integration by parts for the last equation. Together with $S_{1}$ this
proves our proposition.

For $j\in{\mathbb{Z}}$ define

$\mathcal{D}^{2}_{j}=(1-j)D+e_{1}D^{2},$ (7)

where $D$ denotes the ordinary differentiation operator.

It is easy to see that

$\mathcal{D}^{2}_{j}=e_{j}D(e_{1-j}D).$ (8)

We mention that $\mathcal{D}_{j}^{2}e_{j}=0$.

###### Theorem 8

Let $f\in E$ such that $\mathcal{D}_{j}^{2}f\in E$. Furthermore we assume that
$f^{\prime}(0)=0$ in the case $j\geq 2$. Then for sufficiently large $n$ the
operators $S_{n,j}$ and $\mathcal{D}_{j}^{2}$ commute, i.e.,

$\left(S_{n,j}\circ\mathcal{D}_{j}^{2}\right)f=\left(\mathcal{D}_{j}^{2}\circ
S_{n,j}\right)f.$

Proof. For $j=1$ see [9, Theorem 3.2] and for $j\leq 1$ [7, Section 5] and the
references therein.

Let $j\geq 2$. From Proposition 5 we have

$\displaystyle\mathcal{D}^{2}_{j}(S_{n,j}f)(x)$ $\displaystyle=$
$\displaystyle(1-j)n\sum_{k=j-1}^{\infty}s_{n,k}(x)\int_{0}^{\infty}s_{n,k+1-j}(t)f^{\prime}(t)dt$
$\displaystyle+nx\sum_{k=j-2}^{\infty}s_{n,k}(x)\int_{0}^{\infty}s_{n,k+2-j}(t)f^{\prime\prime}(t)dt.$

Now we apply the operator $S_{n,j}$ to $\mathcal{D}^{2}_{j}f$, i.e.,

$\displaystyle(S_{n,j}\mathcal{D}^{2}_{j}f)(x)$ $\displaystyle=$
$\displaystyle(1-j)n\sum_{k=j}^{\infty}s_{n,k}(x)\int_{0}^{\infty}s_{n,k-j}(t)f^{\prime}(t)dt$
$\displaystyle+n\sum_{k=j}^{\infty}s_{n,k}(x)\int_{0}^{\infty}s_{n,k-j}(t)tf^{\prime\prime}(t)dt$
$\displaystyle=:$ $\displaystyle T_{1}+T_{2}$

Using $nts_{n,k-j}(t)=(k+1-j)s_{n,k+1-j}(t)$ and $ks_{n,k}(x)=nxs_{n,k-1}(x)$
we get

$\displaystyle T_{2}$ $\displaystyle=$ $\displaystyle
nx\sum_{k=j}^{\infty}s_{n,k-1}(x)\int_{0}^{\infty}s_{n,k+1-j}(t)f^{\prime\prime}(t)dt$
$\displaystyle+(1-j)\sum_{k=j}^{\infty}s_{n,k}(x)\int_{0}^{\infty}s_{n,k+1-j}(t)f^{\prime\prime}(t)dt.$

Index transform in the first sum, integration by parts in the second sum and
application of $s_{n,k+1-j}^{\prime}(t)=n(s_{n,k-j}-s_{n,k+1-j})$ leads to

$\displaystyle T_{2}$ $\displaystyle=$ $\displaystyle
nx\sum_{k=j-1}^{\infty}s_{n,k}(x)\int_{0}^{\infty}s_{n,k+2-j}(t)f^{\prime\prime}(t)dt$
$\displaystyle+(1-j)n\sum_{k=j}^{\infty}s_{n,k}(x)\int_{0}^{\infty}s_{n,k+1-j}(t)f^{\prime}(t)dt$
$\displaystyle-(1-j)n\sum_{k=j}^{\infty}s_{n,k}(x)\int_{0}^{\infty}s_{n,k-j}(t)f^{\prime}(t)dt.$

Thus

$\displaystyle\mathcal{D}^{2}_{j}(S_{n,j}f)(x)-(S_{n,j}\mathcal{D}^{2}_{j}f)(x)$
$\displaystyle=$
$\displaystyle(1-j)ns_{n,j-1}(x)\int_{0}^{\infty}s_{n,0}(t)f^{\prime}(t)dt+nxs_{n,j-2}(x)\int_{0}^{\infty}s_{n,0}(t)f^{\prime\prime}(t)dt$
$\displaystyle=$ $\displaystyle 0$

as $nxs_{n,j-2}(x)=(j-1)s_{n,j-1}(x)$ and
$\int_{0}^{\infty}s_{n,0}(t)f^{\prime\prime}(t)dt=n\int_{0}^{\infty}s_{n,0}(t)f^{\prime}(t)dt$.

Define

$\mathcal{D}^{2l}_{j}:=e_{j}D^{l}(e_{l-j}D^{l}),\,l\in{\mathbb{N}}.$ (9)

For $l=1$ this is in agreement with (7). Moreover, define $D^{0}_{j}$ to be
the identity operator.

###### Proposition 9

For $l\in{\mathbb{N}}$ the differential operator $\mathcal{D}^{2l}_{j}$ can be
written as $l$-th iterate of $\mathcal{D}^{2}_{j}$, i.e.,

$\mathcal{D}_{j}^{2l}=(\mathcal{D}_{j}^{2})^{l}.$ (10)

Proof. For $j\leq 0$ see [7, (13)]. We prove the relation (10) by induction on
$l$.

For $l=1$ it is sufficient to use (7).

Suppose (10) is true for $l$. Then

$\displaystyle\mathcal{D}^{2l}_{j}\circ\mathcal{D}^{2}_{j}$ $\displaystyle=$
$\displaystyle\mathcal{D}^{2l}_{j}[(1-j)D+e_{1}D^{2}]$ $\displaystyle=$
$\displaystyle e_{j}D^{l}\left\\{e_{l-j}D^{l}[(1-j)D+e_{1}D^{2}]\right\\}$
$\displaystyle=$ $\displaystyle
e_{j}D^{l}\left\\{e_{l-j}[(l+1-j)D^{l+1}+e_{1}D^{l+2}]\right\\}$

and

$\displaystyle\mathcal{D}^{2l+2}_{j}$ $\displaystyle=$ $\displaystyle
e_{j}D^{l+1}[(l+1-j)e_{l-j}D^{l+1}+e_{l+1-j}D^{l+2}]$ $\displaystyle=$
$\displaystyle
e_{j}D^{l}\left\\{e_{l-j}[(l+1-j)D^{l+1}+e_{1}D^{l+2}]\right\\}.$

As a corollary of Proposition 9 we derive an explicit representation of the
differential operator.

###### Corollary 10

For $l\in\mathbb{N}$ we have

${\mathcal{D}}^{2l}_{j}=({\mathcal{D}}^{2}_{j})^{l}=\sum_{i=0}^{l}\binom{l}{i}\left(l-j\right)^{\underline{l-i}}e_{i}D^{l+i}.$

Proof. According to (9), we have

$\displaystyle\mathcal{D}^{2l}_{j}$ $\displaystyle=$ $\displaystyle
e_{j}D^{l}(e_{l-j}D^{l})$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{l}\binom{l}{i}\left(l-j\right)^{\underline{l-i}}e_{i}D^{l+i}.$

Observe that
${\mathcal{D}}^{2l}_{j}=\sum_{i=j}^{l}\binom{l}{i}\left(l-j\right)^{\underline{l-i}}e_{i}D^{l+i}$
in case of $l\geq j\geq 0$.

##  4 Eigenstructure of $S_{n,j}$

In this section we study the eigenstructure of the operators $S_{n,j}$,
$j\in\mathbb{Z}$. As we already know, $e_{0}$ is an eigenfunction with
eigenvalue $1$ for each $j\in\mathbb{Z}$. Now we present a family of
eigenfunctions $g_{j,p}$, $p\in\mathbb{R}$, of $S_{n,j}$ which are independent
of $n$ and can be expressed as modified Bessel functions $I_{j}$ of the first
kind.

###### Theorem 11

Let $j\in\mathbb{Z}$. For each $p\in{\mathbb{R}}$, the function

$\displaystyle g_{j,p}(x)$
$\displaystyle=\sum_{m=\max{\\{0,-j\\}}}^{\infty}\frac{p^{m}x^{m+j}}{m!(m+j)!}$

is an eigenfunction of $S_{n,j}$ with eigenvalue $e^{p/n}$.

Proof. For $p=0$ and $j\geq 0$ we get
$g_{j,0}(x)=\frac{x^{j}}{j!}=\frac{1}{j!}S_{n,j}(e_{j};x)$ by the preservation
of $e_{j}$.

Now let $p\not=0$.

By using (2) we get for $j\in\mathbb{N}_{0}$

$\displaystyle(S_{n,j}g_{j,p})(x)$ $\displaystyle=$
$\displaystyle\sum_{k=j}^{\infty}s_{n,k}(x)\sum_{m=0}^{\infty}\frac{\left(\frac{p}{n}\right)^{m}}{m!(m+j)!}\cdot\frac{(k+m)!}{(k-j)!}n^{-j}$
$\displaystyle=$ $\displaystyle
e^{-nx}\sum_{m=0}^{\infty}\frac{\left(\frac{p}{n}\right)^{m}}{m!(m+j)!}\sum_{k=0}^{\infty}\frac{n^{k}x^{k+j}}{(k+j)!}\cdot\frac{(k+m+j)!}{k!}.$

As

$\displaystyle\sum_{k=0}^{\infty}\frac{n^{k}x^{k+j}}{(k+j)!}\cdot\frac{(k+m+j)!}{k!}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{n^{k}}{k!}\left(x^{k+m+j}\right)^{(m)}$
$\displaystyle=\frac{d^{m}}{dx^{m}}\left(x^{m+j}e^{nx}\right)$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{m}\binom{m}{l}\frac{(m+j)!}{(l+j)!}x^{l+j}n^{l}e^{nx},$

we derive

$\displaystyle(S_{n,j}g_{j,p})(x)$ $\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}\left(\frac{p}{n}\right)^{m}\sum_{l=0}^{m}\frac{n^{l}x^{l+j}}{l!(l+j)!(m-l)!}$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\frac{x^{l+j}p^{l}}{l!(l+j)!}\sum_{m=l}^{\infty}\frac{\left(\frac{p}{n}\right)^{m-l}}{(m-l)!}$
$\displaystyle=$ $\displaystyle e^{\frac{p}{n}}g_{j,p}(x).$

For $-j\in\mathbb{N}$ we get in a similar way

$\displaystyle(S_{n,j}g_{j,p})(x)$ $\displaystyle=$ $\displaystyle
e^{-nx}\sum_{m=-j}^{\infty}\frac{\left(\frac{p}{n}\right)^{m}n^{-j}x^{j}}{m!(m+j)!}\sum_{k=0}^{\infty}\frac{n^{k}x^{k-j}}{(k-j)!}\cdot\frac{(k+m)!}{k!}.$

As

$\displaystyle\sum_{k=0}^{\infty}\frac{n^{k}x^{k-j}}{(k-j)!}\cdot\frac{(k+m)!}{k!}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{n^{k}}{k!}\left(x^{k+m}\right)^{(m+j)}$
$\displaystyle=\frac{d^{m+j}}{dx^{m+j}}\left(x^{m}e^{nx}\right)$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{m+j}\binom{m+j}{l}\frac{m!}{(l-j)!}x^{l-j}n^{l}e^{nx},$

we conclude

$\displaystyle S_{n,j}(g_{j,p};x)$ $\displaystyle=$
$\displaystyle\sum_{m=-j}^{\infty}\left(\frac{p}{n}\right)^{m}\sum_{l=0}^{m+j}\frac{n^{l-j}x^{l}}{l!(l-j)!(m+j-l)!}$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\frac{x^{l}p^{l-j}}{l!(l-j)!}\sum_{m=l-j}^{\infty}\frac{\left(\frac{p}{n}\right)^{m+j-l}}{(m+j-l)!}$
$\displaystyle=$ $\displaystyle e^{\frac{p}{n}}g_{j,p}(x).$

###### Remark 12

For $p\not=0$ the eigenfunctions can be written in terms of modified Bessel
functions of the first kind, i.e.,

$g_{j,p}(x)=(px)^{\frac{j}{2}}I_{j}(2\sqrt{px}).$ (11)

##  5 Eigenstructure of $\mathcal{D}_{j}^{2}$

The Bessel differential equation is given by

$z^{2}y^{\prime\prime}+zy^{\prime}+\left(z^{2}-\nu^{2}\right)y=0.$ (12)

Solutions are the Bessel functions of the first kind $J_{\nu}\left(z\right)$
and of the second kind $Y_{\nu}\left(z\right)$ (also called Weber’s function).
Each is a holomorphic function of $z$ throughout the $z$-plane cut along the
negative real axis. $J_{\nu}\left(z\right)$ and $J_{-\nu}\left(z\right)$ are
linearly independent except when $\nu$ is an integer. In our case we consider
the integer $\nu=j$. Therefore, we take as a fundamental system the functions
$J_{\nu}\left(z\right)$ and $Y_{\nu}\left(z\right)$ which are linearly
independent for all values of $\nu$. For integer $\nu$, $J_{\nu}$ is an entire
function with series expansion

$J_{\nu}\left(z\right)=\left(-1\right)^{\nu}J_{-\nu}\left(z\right)=\left(\frac{z}{2}\right)^{\nu}\sum_{k=0}^{\infty}\frac{\left(-\frac{1}{4}z^{2}\right)^{k}}{k!\Gamma\left(\nu+k+1\right)}\text{
\qquad}\left(z\in C,\text{ }\nu=0,1,2,\ldots\right).$

$Y_{\nu}\left(z\right)$ has a more complicated representation.

We show that the solutions of (12) are closely connected to the eigenfunctions
of the differential operator $\mathcal{D}_{j}^{2}=(1-j)D+e_{1}D^{2}$ as
defined in (7). Since the second order linear ODE $\mathcal{D}_{j}^{2}y=0$ has
the both independent solutions $e_{0},e_{j}$, the kernel of
$\mathcal{D}_{j}^{2}$ is the span of $\left\\{e_{0},e_{j}\right\\}$. The case
$p\neq 0$ will be considered in the next theorem.

###### Theorem 13

Let $j\in\mathbb{Z}$ and $p\in\mathbb{R\smallsetminus}\left\\{0\right\\}$. If
$y\left(x\right)$ is a solution of the Bessel differential equation

$x^{2}y^{\prime\prime}+xy^{\prime}+\left(x^{2}-j^{2}\right)y=0$ (13)

on $\left(0,\infty\right)$, then the function
$h\left(x\right)=g\left(px\right)$, where

$g\left(x\right)=x^{j/2}y\left(2i\sqrt{x}\right)$

is an eigenfunction of $\mathcal{D}_{j}^{2}$ to the eigenvalue $p$.

###### Remark 14

If we take $y\left(x\right)=J_{j}\left(x\right)$ with $j,x\geq 0$, we have

$\displaystyle
h\left(x\right)=\left(px\right)^{j/2}J_{j}\left(2i\sqrt{px}\right)$
$\displaystyle=$
$\displaystyle\left(px\right)^{j/2}\left(i\sqrt{px}\right)^{j}\sum_{k=0}^{\infty}\frac{\left(px\right)^{k}}{k!\Gamma\left(j+k+1\right)}$
$\displaystyle=$ $\displaystyle
i^{j}\left(px\right)^{j}\sum_{k=0}^{\infty}\frac{\left(px\right)^{k}}{k!\Gamma\left(j+k+1\right)}.$

For $p\geq 0$, we essentially have the modified Bessel function $I_{j}$. To
avoid this, we can replace $p$ with $-p$. For negative integers $j$ we can
take advantage of the relation
$J_{j}\left(x\right)=\left(-1\right)^{j}J_{-j}\left(x\right)$.

Proof of Theorem 13. Direct calculation yields

$\displaystyle g^{\prime}\left(x\right)$ $\displaystyle=$
$\displaystyle\frac{j}{2}x^{j/2-1}y\left(2i\sqrt{x}\right)+ix^{\left(j-1\right)/2}y^{\prime}\left(2i\sqrt{x}\right),$
$\displaystyle g^{\prime\prime}\left(x\right)$ $\displaystyle=$
$\displaystyle\frac{j}{2}\frac{j-2}{2}x^{j/2-2}y\left(2i\sqrt{x}\right)+\frac{j}{2}x^{j/2-1}ix^{-1/2}y^{\prime}\left(2i\sqrt{x}\right)$
$\displaystyle+i\frac{j-1}{2}x^{\left(j-3\right)/2}y^{\prime}\left(2i\sqrt{x}\right)+i^{2}x^{\left(j-2\right)/2}y^{\prime\prime}\left(2i\sqrt{x}\right)$
$\displaystyle=$
$\displaystyle\frac{j}{2}\frac{j-2}{2}x^{j/2-2}y\left(2i\sqrt{x}\right)+i\frac{2j-1}{2}x^{\left(j-3\right)/2}y^{\prime}\left(2i\sqrt{x}\right)-x^{\left(j-2\right)/2}y^{\prime\prime}\left(2i\sqrt{x}\right).$

By the Bessel differential equation (13), we have

$y^{\prime\prime}\left(2i\sqrt{x}\right)=-\frac{1}{2i\sqrt{x}}y^{\prime}\left(2i\sqrt{x}\right)-\left(1+\frac{j^{2}}{4x}\right)y\left(2i\sqrt{x}\right)$

and it follows that

$\displaystyle g^{\prime\prime}\left(x\right)$ $\displaystyle=$
$\displaystyle\frac{j^{2}-2j}{4}x^{j/2-2}y\left(2i\sqrt{x}\right)+i\frac{2j-1}{2}x^{\left(j-3\right)/2}y^{\prime}\left(2i\sqrt{x}\right)$
$\displaystyle-\frac{i}{2}x^{\left(j-3\right)/2}y^{\prime}\left(2i\sqrt{x}\right)+\left(x^{j/2-1}+\frac{j^{2}}{4}x^{j/2-2}\right)y\left(2i\sqrt{x}\right)$
$\displaystyle=$
$\displaystyle\left(x^{j/2-1}+\frac{j^{2}-j}{2}x^{j/2-2}\right)y\left(2i\sqrt{x}\right)+i\left(j-1\right)x^{\left(j-3\right)/2}y^{\prime}\left(2i\sqrt{x}\right).$

Hence,

$\displaystyle
p^{-1}\left(p^{2}xg^{\prime\prime}\left(px\right)+\left(1-j\right)pg^{\prime}\left(px\right)\right)$
$\displaystyle=$
$\displaystyle\left(\left(px\right)^{j/2}+\frac{j^{2}-j}{2}\left(px\right)^{j/2-1}\right)y\left(2i\sqrt{px}\right)+i\left(j-1\right)\left(px\right)^{\left(j-1\right)/2}y^{\prime}\left(2i\sqrt{px}\right)$
$\displaystyle+\frac{j\left(1-j\right)}{2}\left(px\right)^{j/2-1}y\left(2i\sqrt{px}\right)+i\left(1-j\right)\left(px\right)^{\left(j-1\right)/2}y^{\prime}\left(2i\sqrt{px}\right)$
$\displaystyle=$
$\displaystyle\left(\left(px\right)^{j/2}+\frac{j^{2}-j}{2}\left(px\right)^{j/2-1}\right)y\left(2i\sqrt{px}\right)+\frac{j\left(1-j\right)}{2}\left(px\right)^{j/2-1}y\left(2i\sqrt{px}\right)$
$\displaystyle=$
$\displaystyle\left(px\right)^{j/2}y\left(2i\sqrt{px}\right)=g\left(px\right),$

i.e., the function $h\left(x\right)=g\left(px\right)$ is a solution of the ODE

$xh^{\prime\prime}+\left(1-j\right)h^{\prime}=ph.$

Noting definition (7) concludes the proof.

The next theorem shows a connection between eigenfunctions of $S_{n,j}$ and
eigenfunctions of $\mathcal{D}_{j}^{2}$.

###### Theorem 15

Let $j\in\mathbb{Z}$ and $p\in\mathbb{R}$. If a function $h\in
C^{2}\left(0,\infty\right)$ is an eigenfunction of $S_{n,j}$ to the eigenvalue
$e^{p/n}$ for infinitely many integers $n$, then $h$ is an eigenfunction of
$\mathcal{D}_{j}^{2}$ to the eigenvalue $p$.

Proof. Since $h\in C^{2}\left(0,\infty\right)$, we have for each $x>0$ the
Voronovskaja relation

$\left(S_{n,j}h\right)\left(x\right)=h\left(x\right)+n^{-1}\left(\mathcal{D}_{j}^{2}h\right)\left(x\right)+o\left(1/n\right)\text{
\qquad}\left(n\rightarrow\infty\right).$

On the other hand, we have

$\left(S_{n,j}h\right)\left(x\right)=e^{p/n}h\left(x\right)=\left(1+\frac{p}{n}+o\left(\frac{1}{n}\right)\right)h\left(x\right)\text{
\qquad}\left(n\rightarrow\infty\right).$

Here $n\rightarrow\infty$ means that we take only such $n$ for which
$S_{n,j}h=e^{p/n}h$. Comparing both relations we infer that

$\left(\mathcal{D}_{j}^{2}h\right)\left(x\right)=ph\left(x\right)+o\left(1\right)\text{
\qquad}\left(n\rightarrow\infty\right),$

which completes the proof.

The converse problem is more complicated. It is easy to show the following
result.

###### Theorem 16

If a function $h\in C^{2}\left(0,\infty\right)\cap E$ is an eigenfunction of
$\mathcal{D}_{j}^{2}$ to the eigenvalue $p$ on $\left(0,\infty\right)$, it
follows, for each $x>0$ and for each constant $c>0$, the asymptotic relation

$\left(S_{n,j}h\right)\left(x\right)=e^{p/n}h\left(x\right)+o\left(n^{-c}\right)\text{
\qquad}\left(n\rightarrow\infty\right).$

Proof. The fact, that $h$ is an eigenfunction of $\mathcal{D}_{j}^{2}$ implies
that $h$ is infinitely often differentiable. Let $q$ be a positive integer. By
[1, Theorem 25] the operators $S_{n,j}$ possess the asymptotic expansion

$\left(S_{n,j}h\right)\left(x\right)=h\left(x\right)+\sum_{k=1}^{q}\frac{1}{k!n^{k}}\left({\mathcal{D}}_{j}^{2k}h\right)\left(x\right)+o\left(n^{-q}\right)\text{
\qquad}\left(n\rightarrow\infty\right).$

Observing that
${\mathcal{D}}_{j}^{2k}h=\left({\mathcal{D}}_{j}\right)^{k}h=p^{k}h$ (see
Proposition 9) and

$\sum_{k=0}^{q}\frac{p^{k}}{k!n^{k}}=e^{p/n}+O\left(n^{-q-1}\right)\text{
\qquad}\left(n\rightarrow\infty\right)$

completes the proof.

## References

  * [1] U. Abel, A. M. Acu, M. Heilmann, I. Raşa, Asymptotic properties for a general class of Szász-Mirakjan-Durrmeyer operators, arXiv:2407.16474.
  * [2] U. Abel, M. Ivan, Enhanced asymptotic approximation and approximation of truncated functions by linear operators, Constructive Theory of Functions, Proc. Int. Conf. CTF, Varna, June 2 - June 6, 2005, B. D. Bojanov (Ed.), Prof. Marin Drinov Academic Publishing House, 2006, 1–10.
  * [3] M. Abramowitz, I. A. Stegun, (Eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards Applied Mathematics Series 55, Issued June 1964, Tenth Printing, December 1972, with corrections.
  * [4] M. Heilmann, Commutativity of operators from Baskakov-Durrmeyer type, Constructive theory of functions (Varna, 1987), Publ. House Bulgar. Acad. Sci., Sofia, pp. 197–206, 1988.
  * [5] M. Heilmann, Direct and converse results for operators of Baskakov-Durrmeyer type, Approximation Theory Appl. 5 (1):105–127 (1989).
  * [6] M. Heilmann, Erhöhung der Konvergenzgeschwindigkeit bei der Approximation von Funktionen mit Hilfe von Linearkombinationen spezieller positiver linearer Operatoren. Habilitationschrift Universität Dortmund, 1991.
  * [7] M. Heilmann, Commutativity and spectral properties of genuine Baskakov-Durrmeyer type operators and their $k$th order Kantorovich modification, J. Numer. Anal. Approx. Theory, 44, No.2, 166–179 (2015).
  * [8] M. Heilmann, M. W. Müller, On simultaneous approximation by the method of Baskakov-Durrmeyer operators, Numer. Funct. Anal. Optim. 10 (1989), 127–138.
  * [9] M. Heilmann, G. Tachev, Commutativity, direct and strong converse results for Philipps operators, East J. Approx. 17 (3), 299–317 (2011).
  * [10] M. Heilmann, G. Tachev, Linear combinations of genuine Szász-Mirakjan-Durrmeyer operators, Advances in Applied Mathematics and Approximation - Contributions from AMAT 2012, Band 41, Seite 85-106, Applied Mathematics and Approximation Theory 2012, Ankara, Turkey. May 2012, In Anastassiou, George A. and Duman, Oktay, Editor, Herausgeber: Springer Proceedings in Mathematics and Statistics, 2013
  * [11] S. M. Mazhar, V. Totik, Approximation by modified Szász operators, Acta Sci. Math. 49, 257–269 (1985).
  * [12] R. S. Phillips, An inversion formula for Laplace transforms and semi-groups of linear operators, Ann. Math. (2) 59 (1954), 325–356.
  * [13] T. Świderski, On certain properties of the combinations of Szász-Durrmeyer operators, Anal. Theory Appl. Vol. 27(2) (2011), 167–180.