# On the controversial turbulent Schmidt number value in particle-laden
boundary layer flows

J. Chauchat  Univ. Grenoble Alpes, CNRS, Grenoble INP*, LEGI, 38000 Grenoble,
France* Institute of Engineering Univ. Grenoble Alpes  D. Hurther  Univ.
Grenoble Alpes, CNRS, Grenoble INP*, LEGI, 38000 Grenoble, France* Institute
of Engineering Univ. Grenoble Alpes  T. Revil-Baudard  Univ. Grenoble Alpes,
CNRS, Grenoble INP*, LEGI, 38000 Grenoble, France* Institute of Engineering
Univ. Grenoble Alpes  Z. Cheng  Civil and Environmental Engineering, Center
for Applied Coastal Research, University of Delaware, Newark, DE 19711, USA
Now at Convergent Science Inc., Madison, WI 53719, USA  T.-J. Hsu  Civil and
Environmental Engineering, Center for Applied Coastal Research, University of
Delaware, Newark, DE 19711, USA<EMAIL_ADDRESS>
###### Abstract

One of the most enigmatic science question concerning inertial particle
transport by a turbulent boundary layer flow is the value of the turbulent
Schmidt number defined as the ratio of turbulent eddy viscosity to particle
concentration diffusivity. Using direct acoustic measurement of turbulent
particle flux profile, and two-phase flow turbulence-resolving numerical
simulation, it is demonstrated that turbulent dispersion of particles is
reduced rather than enhanced as predicted by many existing literature models.
The explanation lies in the misleading assumption of settling velocity in
quiescent water to estimate the turbulent particle diffusivity, while direct
measurements and simulations of turbulent particle flux support the occurrence
of settling retardation. The analysis presented herein suggests that the value
of the turbulent Schmidt number is always larger than unity with values
between 3 and 4 based on the directly measured turbulent particle flux. The
observed settling reduction can not be explained by the well-known hindrance
effects related to particle concentration. This effect seems to be related to
turbulence-particle interactions and correlates more with the Stokes number.
At last, the new parameters, namely the turbulent Schmidt number higher than
unity, modified von Kármán constant, and settling retardation, are
successfully tested for the modeling of particle concentration profile using
the well-known Rouse formulation. This result suggests that new
parametrizations are possible to reduce the degree of empiricism to predict
suspended particle transport by a boundary layer flow.

††preprint: APS/123-QED

##  I Introduction

The transport of solid particles by a turbulent fluid flow is a key process in
geophysical and industrial two-phase flows such as material and food
processing, pneumatic transport, fluidized beds, slurry flows or sediment
transport. In all these situations, interactions between particles and fluid
turbulence are key physical processes that contribute to the dynamics of the
system. Among the consequences induced by the presence of particles, the
modification of fluid turbulence and the turbulent dispersion of particles are
crucial mechanisms that are not fully understood Balachandar and Eaton (2010).
The complexity arises when particles are inertial, i.e. when the particle
response time $\tau_{p}$ is larger than a representative turbulent eddy time-
scale $\tau_{f}$, leading to Stokes number ${\rm St}=\tau_{p}/\tau_{f}$
greater than unity. Inertial particles act as a local filter for the turbulent
kinetic energy spectrum due to their not fully-correlated movements with
turbulent eddies of similar or smaller size than the particle itself. This
problem has been studied in many different flow configurations including
Homogeneous Isotropic Turbulence, jets and turbulent wall-bounded flows.
However, further complexity arises under gravitational acceleration.

In sediment transport, under intense flows corresponding to high fluid bed
shear stress relative to the buoyant weight of the particles, the so-called
Shields number $\theta$, strong erosions from the underlying sediment bed are
observed and a significant amount of particles are transported in suspension
by fluid turbulent eddies Garcia (2008). The coupling between the sediment
flux and the underlying bed topography controls the large-scale morphological
evolution of rivers, estuaries and coastal nearshore zones Aagaard and Hughes
(2010). In most of these situations, the transported particles are made of
inorganic non-cohesive sand grains with Stokes number values well above unity,
i.e. inertial particles. Sediment transport is a truly multi-scale process
coupling fine scale turbulence-particle interactions (O($\mu m$) - O(ms)) with
the morphological evolution of natural systems at large scales (O($km$) -
O(yrs)). Therefore, elucidation of the governing interaction processes
associated with the transport of inertial particles by a turbulent boundary
layer flow is a major scientific issue.

The quantity of particular interest is the stream-wise particle flux defined
as the integral over the flow depth of the product between the local particle
velocity and particle concentration. Consequently, it is necessary to
understand the key processes that controls the velocity and concentration
profiles. On the one hand, the presence of particles may significantly modify
turbulence (e.g. Vanoni, 1975; Kiger and Pan, 2002; Revil-Baudard  _et al._ ,
2015; Cheng  _et al._ , 2018). This is often characterized by a modification
of the von Kármán constant $\kappa$ in the classical logarithmic law-of-the-
wall. On the other hand, the turbulent diffusivity of particle concentration
$\epsilon_{\phi}$ may significantly differ from the fluid eddy-viscosity
$\nu^{t}$ Lyn (2008). These two quantities are the key ingredients to predict
the particle flux. Under fully-developped turbulent flow conditions and by
assuming a local balance between the gravity-driven settling flux
$W_{s}\langle\phi\rangle$ and the upward Reynolds particle flux $\langle
w^{\prime}\phi^{\prime}\rangle$, the particle phase mass conservation equation
reduces to:

$-W_{s}\langle\phi\rangle+\langle w^{\prime}\phi^{\prime}\rangle=0.$ (1)

In this equation, $W_{s}$ is the particle settling velocity and $\langle
w^{\prime}\phi^{\prime}\rangle$ is the wall-normal Reynolds particle flux. In
the later, $w^{\prime}$ stands for the velocity fluctuations and
$\phi^{\prime}$ for the concentration fluctuations defined with respect to
their respective mean values where $\langle\rangle$ represents a temporal
averaging operator. Rouse (1938) proposed to model the Reynolds particle flux
using a Fickian gradient diffusion model:

$\langle
w^{\prime}\phi^{\prime}\rangle=-\epsilon_{\phi}\dfrac{d\langle\phi\rangle}{dz},$
(2)

where $\epsilon_{\phi}$, the particle concentration turbulent diffusivity, is
linked to the eddy-viscosity through the turbulent Schmidt number defined as
${\rm S_{c}}=\nu^{t}/\epsilon_{\phi}$. Using a Prandtl’s mixing length model
with a free surface correction for the eddy-viscosity $l_{m}=\kappa
z(1-z/H_{f})^{1/2}$, where $\kappa$ is the von Kármán constant, $z$ is the
wall-normal distance and $H_{f}$ is the water depth above the particle bed and
by further assuming that particle settling velocity $W_{s}$ can be
approximated by its value in quiescent water $W^{0}_{s}$, Rouse (1938) derived
an analytical solution for the mean particle concentration profile as:

$\dfrac{\phi}{\phi_{r}}=\left(\dfrac{H_{f}-z}{z}\dfrac{z_{r}}{H_{f}-z_{r}}\right)^{Ro},$
(3)

where $\phi_{r}=\phi(z_{r})$ is a reference concentration taken at an
elevation $z_{r}$ above the particle bed and located within the suspension
layer, i.e. where $\phi<0.08$, and ${\rm Ro=S_{c}}W^{0}_{s}/\kappa u_{*}$ is
the Rouse number corresponding to the slope of the particle concentration
profile. In this definition, $u_{*}$ represents the bed friction velocity.
Using measured concentration profile and the estimated bed friction velocity
and particle settling velocity, the Schmidt number can be readily estimated
using the Rouse profile.

Discrepancies between a measured concentration profile and the Rouse profile
have been attributed to a modification of the particle diffusivity compared
with the eddy viscosity, leading to values of the turbulent Schmidt number
${\rm S_{c}}$ different from unity. Because this classical method estimates
the Schmidt number by best fitting the measured concentration profile with the
Rouse profile and leaving ${\rm S_{c}}$ as the only free parameter, the
resulting ${\rm S_{c}}$ is a depth-averaged value noted as
${\rm\overline{S_{c}}}$ in the following. Figure 1a summarizes several
experimental values reported in the literature for ${\rm\overline{S_{c}}}$
plotted as a function of $W_{s}/u_{*}$ Lyn (2008). The first observation that
can be made is the large scatter of the data with values lower and higher than
unity. Values lower (higher) than unity can be interpreted as an enhanced
(reduced) dispersion of particles compared to fluid momentum. Van Rijn (1984)
suggested that turbulent dispersion of inertial particles is enhanced due to
centrifugal force which tends to throw particles out of small-scale turbulent
vortexes and proposed an empirical model written as:

${\rm\overline{S_{c}}}=\left[1+2\left(\dfrac{W_{s}^{0}}{u_{*}}\right)^{2}\right]^{-1}.$
(4)

While this model (solid line in Figure 1a) is widely used, it does not match
the laboratory data of Lyn (1988).

The turbulent Schmidt number can also be obtained locally provided that the
concentration profile is measured with enough accuracy to resolve the sharp
wall-normal gradients near the flow bed. Figure 1b shows a summary of existing
data for locally estimated Schmidt number as a function of the normalized
depth $z/H_{f}$. Again ${\rm S_{c}}$ shows large scatter with values ranging
from as low as 0.2 to as large as 2.5. In the experiments reported by Lyn
(1988) and Barton and Lin (1955), only the mean concentration profile was
measured and so ${\rm S_{c}}$ is obtained by using both the mass balance and
the gradient diffusion model (Eqs. 1-2) and therefore under the assumption
that $W_{s}=W_{s}^{0}$. In Cellino and Graf (1999), using the novel acoustic
particle flux profiler of Shen and Lemmin (1999), the authors measured
directly the turbulent particle flux and hence only Eq. 2 is used to calculate
${\rm S_{c}}$ directly. The mean concentration profile used to estimate ${\rm
S_{c}}$ was obtained from an iso-kinetic flow suction system with a much lower
vertical resolution than the direct measurement of the Reynolds particle flux
Cellino (1998). This introduces a source of uncertainty in their results
especially in the near-bed region where the concentration gradient is very
high. Despite this source of measurement uncertainty, it is interesting to
point out that almost all their data (circles in 1b) suggest ${\rm S_{c}}>1$.

The potential reasons for the discrepancies in the Schmidt number values
reported in the literature are (i) the assumption of using particle settling
velocity in quiescent water and (ii) the assumption of the von Kármán constant
value being identical to the clear water value ($\kappa=0.41$). Concerning the
first assumption, several publications have reported that the settling
velocity of individual particles could be modified in turbulent flows Murray
(1970); Good  _et al._ (2014); Cuthbertson and Ervine (2007); Kawanisi and
Shiozaki (2008); Zhou and Cheng (2009); Akutina  _et al._ (2020); Mora  _et
al._ (2021). However, no consensus has been reached on the effect of
turbulence on the settling velocity of individual particles as some studies
have found settling velocity enhancement while others observed settling
retardation (e.g. Nielsen, 1992; Kawanisi and Shiozaki, 2008; Dey  _et al._ ,
2019). Regarding the second assumption, in particle-laden flows, von Kármán
constant reduction has been reported (e.g. Vanoni, 1941; Revil-Baudard  _et
al._ , 2015, 2016). Two physical mechanisms have been proposed to explain this
reduction, density stratification Vilaret and Davies (1995) and/or turbulent
drag work Hsu  _et al._ (2003); Cheng  _et al._ (2018). The fact that the
ratio ${\rm S_{c}}/\kappa$ appears in the Rouse number definition could
explain some of the observed discrepancies. In order to address this question,
it is mandatory to measure concurrently and with sufficient accuracy the
velocity and the concentration profiles to allow for a direct estimate of the
von Kármán constant and of the turbulent Schmidt number. This ability
represents a major measurement challenge under such energetic particle-laden
flow conditions.

To the best of our knowledge, the first measurements reported in the
literature showing turbulent fluxes, mean concentration and velocity profiles
are those of Shen and Lemmin (1999) using the Acoustic Particle Flux Profiler.
Over the past ten years, we have intensively developed and improved this
technology as the Acoustic Concentration and Velocity Profiler (ACVP) offering
a unique wide-band multi-frequency capability for simultaneous vertical
profiles of velocity and particle concentration at turbulence resolving scales
($\Delta z\approx 3$ mm ; $f=78$ Hz) Hurther and Thorne (2011); Hurther  _et
al._ (2011); Thorne and Hurther (2014); Naqshband  _et al._ (2017); Fromant
_et al._ (2018, 2019). We successfully applied this technology to study
intense sediment transport processes in the so-called sheet-flow regime in
Revil-Baudard  _et al._ (2015). This unique data-set allows for the first time
to investigate the relationship between Reynolds particle flux, Reynolds
stresses, and velocity and particle concentration gradients in order to shed
new light on the turbulent Schmidt number value.

Over the last two decades, significant progress has been made on the numerical
modeling of sediment transport by using two-phase flow approaches (e.g.
Chauchat and Guillou, 2008; Jha and Bombardelli, 2010; Revil-Baudard and
Chauchat, 2013; Lee  _et al._ , 2016; Cheng  _et al._ , 2017; Chauchat, 2018).
Recently, we have performed the first turbulence-resolving two-fluid
simulation Cheng  _et al._ (2018) that has been validated against experimental
data from Revil-Baudard  _et al._ (2015). In this approach, the most energetic
turbulent flow scales are directly resolved as well as the particle dynamics
allowing the majority of turbulent particle fluxes to be resolved.

In this contribution, using the recent high-resolution experimental data-set
of Revil-Baudard  _et al._ (2015) and turbulence-resolving two-phase flow
simulations of Cheng  _et al._ (2018), we are able to provide a physical
explanation for the contradictory findings in the literature.

Figure 1: Experimental data for the turbulent Schmidt number ${\rm S_{c}}$
from Lyn (2008): a) Averaged values of ${\rm S_{c}}$ over $0.1<z_{/}H_{f}<0.5$
as a function of $W_{s}/u_{*}$: $\triangle$ Barton and Lin (1955), $+$ Coleman
(1970), $\blacksquare$ Lyn (1988), the black solid line corresponds to Van
Rijn (1984) model (Eq. 4) ; b) Local estimates of ${\rm S_{c}}$ for uniform
flow over a plane equilibrium bed as a function of relative distance,
$z/H_{f}$, from the bed: $\circ$ Cellino and Graf (1999), $\blacksquare$ Lyn
(1988), $\blacktriangle$ Lyn (1988), $\square$ Barton and Lin (1955),
$\triangle$ Barton and Lin (1955).

##  II Material and method

###  II.1 Sediment transport experiments

In Revil-Baudard  _et al._ (2015) sediment transport experiments have been
carried out in the LEGI/ENSE3 tilting flume. The mobile bed experiment is
denoted as $RB15$ and corresponds to a Shields number of
$\theta=u_{*}^{2}/[(\rho_{p}/\rho_{f}-1)gd_{p}]=0.4$ where $\rho^{f}$ and
$\rho^{s}$ are fluid and particle densities, $g$ is gravity acceleration and
$d_{p}$ is the particle diameter ; and a suspension number of
$W_{s}^{0}/u_{*}=1.2$ (see table 1). For this experiment the slope has been
set to $S_{0}=0.5\%$. The experimental condition has been realized $N=11$
times for $RB15$ in order to verify the reproducibility of the experiment and
to perform ensemble averaging. The experimental parameters are presented in
table 1.

The ACVP technology was used to measure the vertical profiles, from the non-
moving particle bed up to the free-surface, of the two velocity components
$u(z),w(z)$ and of the particle volumetric concentration $\phi(z)$ with
vertical spatial resolution of $\Delta z=3$mm at 78 Hz (see Thorne and Hurther
(2014) for details on acoustic measurement method and Revil-Baudard  _et al._
(2015, 2016) for details on the specific ACVP settings applied in the datasets
used herein).

Table 1: Summary of experimental parameters. Particle diameter ($d_{p}$), density ratio ($\rho_{p}/\rho_{f}$), settling velocity ($W_{s}^{0}$), friction velocity ($u_{*}$), suspension number ($W_{s}^{0}/u_{*}$), bulk Reynolds number (${\rm Re_{b}}=UH_{f}/\nu^{f}$), with $\nu^{f}=10^{-6}m^{2}/s$, particulate Reynolds number (${\rm Re_{p}}=W_{s}^{0}d_{p}/\nu^{f}$), Stokes number based on the Kolmogorov time scale (${\rm\overline{St_{\eta}}}=\tau_{p}/\overline{\tau_{\eta}}$ with $\tau_{p}$ the particle response time and $\overline{\tau_{\eta}}=\overline{\eta}^{2}/\nu^{f}$ the Kolmogorov time-scale with $\overline{\eta}=((\nu^{f}/u_{*})^{3}d_{p})^{1/4}$) and Stokes number based on the viscous time scale (${\rm St^{+}}=\tau_{p}/\tau_{v}$ with $\tau_{v}=\nu^{f}/u_{*}^{2}$) .  Name | Reference | $d_{p}$ | $\rho^{s}/\rho^{f}$ | $W_{s}^{0}$ | $u_{*}$ | $W_{s}/u_{*}$ | ${\rm Re_{b}}$ | ${\rm Re_{p}}$ | ${\rm\overline{St_{\eta}}}$ | ${\rm St^{+}}$
---|---|---|---|---|---|---|---|---|---|---
|  | ($mm$) |  | ($cm/s$) | ($cm/s$) |  | ($\times 10^{4}$) |  |  |
$RB15$ | Revil-Baudard  _et al._ (2015) | $3$ | $1.19$ | $6.2$ | $5.0$ | $1.2$ | $9$ | 186 | 17.4 | 178.8
$SL99\\_1$ | Shen and Lemmin (1999) | $0.13$ | $2.65$ | $1.2$ | $4.8$ | $0.25$ | $26$ | 1.3 | 1.9 | 2.3
$SL99\\_2$ | Shen and Lemmin (1999) | $0.13$ | $2.65$ | $1.2$ | $5.2$ | $0.23$ | $29$ | 1.3 | 2.2 | 2.7

A specific experimental protocol has been applied to obtain steady and uniform
flow conditions in the absence of a particle recirculating facility. First,
the time period $t\in[t_{1}-t_{2}]$ for which the flow is uniform is
identified based on the vertical linearity of the mean turbulent shear-stress
profile. Then the local and instantaneous velocity, concentration and particle
flux measurements are temporally averaged over $\Delta t=t_{2}-t_{1}$ and over
the $N$ realizations of the same experiment to guarantee statistically
converged velocities, concentration and particle flux profiles.

$\displaystyle\langle
A\rangle^{S}(z)=\frac{1}{N}\sum_{i=1}^{N}\left(\frac{1}{\Delta
t}\int_{t_{1}}^{t_{2}}A_{i}(t,z)dt\right).$ (5)

The operator $\langle-\rangle^{S}$ refers to the superficial averaging, and
$A$ is a given measured quantity. The averaging time window is $\Delta t=6$s.

In order to discuss and confirm the validity of the results obtained for RB15
experiments, the experimental dataset of Shen and Lemmin (1999) will be used
for comparison and referred to as runs SL99_1 and SL99_2 in the following. To
the best of our knowledge, it is the only existing experimental dataset
providing the required direct measurements of time-resolved vertical profiles
of two components velocity, particle concentration and fluxes in similar type
of particle-laden flows. The SL99 experiments were carried out in similar
open-channel flow conditions regarding the hydrodynamic regimes (fully
turbulent, fully hydraulically rough, uniform, steady and subcritical flows).
Run SL99_2 is somewhat more energetic than SL99_1 in terms of hydraulic power
but similar regarding the hydrodynamic and sediment transport regimes. The
particles (sand grains of well-sorted unimodal size distribution) used in the
two SL99 experiments explain the important differences seen in Table I for the
suspension number, particle Reynolds and Stokes numbers values. These values
reveal a fully suspension dominated particle-laden flow (with negligible
bedload transport). This is due to the less inertial sand grains having much
smaller size but higher density compared with the PMMA particles used in RB15.
Finally, another important point is that for all considered particle-laden
flows, transport capacity was fully established. This implies that
concentration effects are at their maximal possible level for all considered
fluid-particle interaction processes.

###  II.2 Two-fluid turbulence-resolving simulations

In the two-fluid turbulence-resolving or Large Eddy Simulations (LES) model,
the large-scale turbulent structures are directly computed from filtered
Navier-Stokes equations while the small-scale motions, occurring at spatial
scales smaller than the filter size, are accounted for using sub-grid
closures. In this numerical approach, both the fluid phase and the particle
phase are modeled as a continuum. A Favre filtering approach is used for the
scale separation, and the filtered continuity equations read as,

$\frac{\partial({1-\hat{\phi}})}{\partial{t}}+\frac{\partial{({1-\hat{\phi}})\hat{u}^{f}_{i}}}{\partial{x_{i}}}=0,$
(6)
$\frac{\partial{\hat{\phi}}}{\partial{t}}+\frac{\partial{\hat{\phi}\hat{u}^{s}_{i}}}{\partial{x_{i}}}=0,$
(7)

where $\hat{\phi}$ is the filtered particle volumetric concentration,
$\hat{u}^{f}_{i},\hat{u}^{s}_{i}$ are the filtered fluid and particle
velocities, and $i=1,2,3$ represents streamwise ($x$), spanwise ($y$) and
vertical ($z$) components, respectively.

The filtered momentum equations for fluid phase and particle phase are written
as:

$\displaystyle\frac{\partial{\rho^{f}}({1-\hat{\phi}})\hat{u}^{f}_{i}}{\partial{t}}+\frac{\partial{{\rho^{f}}({1-\hat{\phi}})\hat{u}^{f}_{i}\hat{u}^{f}_{j}}}{\partial{x_{j}}}$
$\displaystyle=$
$\displaystyle-(1-\hat{\phi})\frac{\partial{\hat{p}^{f}}}{\partial{x_{i}}}+\frac{\partial{(1-\hat{\phi})(\hat{\tau}^{f}_{ij}+\hat{\tau}^{f,sgs}_{ij})}}{\partial{x_{j}}}$
(8) $\displaystyle+\rho^{f}(1-\hat{\phi})g_{i}+\hat{M}^{fs}_{i},$
$\frac{\partial{\rho^{s}}\hat{\phi}\hat{u}^{s}_{i}}{\partial{t}}+\frac{\partial{{\rho_{p}}\hat{\phi}\hat{u}^{s}_{i}\hat{u}^{s}_{j}}}{\partial{x_{j}}}=-\hat{\phi}\frac{\partial{\hat{p}^{f}}}{\partial{x_{i}}}+\frac{\partial{\hat{\phi}\hat{\tau}^{s,sgs}_{ij}}}{\partial{x_{j}}}-\frac{\partial{\hat{p}^{s}}}{\partial{x_{i}}}+\frac{\partial{\hat{\tau}^{s}_{ij}}}{\partial{x_{j}}}+\rho^{s}\hat{\phi}g_{i}-\hat{M}^{fs}_{i}$
(9)

where $g_{i}$ is the gravitational acceleration and $\hat{p}^{f}$ is the fluid
pressure. $\hat{\tau}^{f}_{ij}$ and $\hat{\tau}^{f,sgs}_{ij}$ are the fluid
(molecular) viscous stress and subgrid stress associated with the unresolved
turbulent motions. In analogy to the fluid phase, the unresolved particle
motions due to turbulence are taken into account by the subgrid stress,
$\hat{\tau}^{s,sgs}_{ij}$. These subgrid stresses are modeled using a dynamic
Smagorinsky subgrid closure (Germano  _et al._ , 1991; Lilly, 1992). The
particle pressure $\hat{p}^{s}$ and particle stress $\hat{\tau}^{s}_{ij}$ due
to intergranular interactions are modeled by the kinetic theory of granular
flows (Ding and Gidaspow, 1990) and phenomenological closure of contact
stresses (Srivastava and Sundaresan, 2003). $\hat{M}^{fs}_{i}$ represents the
filtered drag force between fluid phase and particle phase, which is only
composed of the resolved part in the present work:

$\hat{M}^{fs}_{i}=-\widehat{\phi\beta{u^{r}_{i}}}\approx-\beta\widehat{{\phi}u^{r}_{i}}=-\beta\hat{\phi}\hat{u}^{r}_{i},$
(10)

where $\hat{u}^{r}_{i}=\hat{u}^{f}_{i}-\hat{u}^{s}_{i}$ is the resolved
relative velocity between fluid phase and particle phase. Contrary to Cheng
_et al._ (2018) the sub-grid drag model is ignored in the present work.

The details of the numerical configuration can be found in Chauchat  _et al._
(2017); Cheng  _et al._ (2018) and the major parameters are summarized in
table 2). Spatial averaging over the two statistically homogeneous $x$ and $y$
directions and time averaging are applied to the simulation results over a
duration of about 60 eddy turn over time $T^{e}=H_{f}/U=0.175$ s to obtain
ensemble-averaged flow statistics.

Table 2: Numerical parameters used for the two-phase flow LES sheet flow simulations. run name | N cells | $\Delta t$ | $\Delta_{x}$ | $\Delta_{y}$ | $\Delta_{z}^{min}$ | $\Delta_{z}^{max}$
---|---|---|---|---|---|---
| (in millions) | (s) | (mm) | (mm) | (mm) | (mm)
$LES$ | 29.2 | $2\times 10^{-4}$ | 1.65 | 1.65 | 0.4 | 2.2

##  III Results and discussion

In the following, our unique high-resolution experimental dataset, denoted as
RB15, and two-fluid turbulence-resolving simulation results, denoted as two-
fluid LES, are analyzed to infer whether ${\rm S_{c}}>1$ or ${\rm S_{c}<1}$
(see table 1).

The averaged profiles of velocity, concentration, Reynolds shear stress and
particle flux corresponding to this configuration are presented in figure 2.
Measured Reynolds stress $\langle u^{\prime}w^{\prime}\rangle$ (symbols in
Figure 2c) peaks near the top of the sheet flow layer (particle volumetric
concentration about 0.08) and shows the expected upward-decaying linear
profile valid for steady, uniform channel flows. Within the sheet flow layer,
Reynolds stress decays sharply towards the bed. Measured particle flux in the
wall-normal direction $\langle w^{\prime}\phi^{\prime}\rangle$ also peaks at
the same vertical location (Figure 2d). The results obtained using two-fluid
LES are compared with measured data from Revil-Baudard  _et al._ (2015) in
figure 2. In terms of velocity and concentration profiles (Figure 2a and b)
the numerical results are in good agreement with the measured data although
particle concentration in the dilute suspension region is underestimated. The
numerical results reproduce measured Reynolds stress $\langle
u^{\prime}w^{\prime}\rangle$ very well. However, the vertical turbulent
particle flux $\langle w^{\prime}\phi^{\prime}\rangle$ is over-predicted
(resp. under-predicted) in the near bed region (resp. in the outer layer).
This probably explains the discrepancies on the concentration profile observed
in the two-fluid LES results. These discrepancies are still an open question.
It is important to recall here that the maximum of the wall-normal turbulent
flux is located in the near-bed region where the particle concentration is
high, i.e. $\phi\in[0.1;0.55]$. At such volumetric concentrations a strong
interplay between turbulence-particle interactions and particle-particle
interactions, the so-called four-way coupling, is taking place. It is highly
possible that some of the closures of the two-fluid model requires
modifications to better reproduce the measurements. This issue deserves
further investigations but more experimental data are needed to guide the
development of the two-fluid model.

Figure 2: Turbulence averaged vertical profiles of mean velocity $\langle
u\rangle$ (a), mean concentration $\langle\phi\rangle$ (b), the Reynolds shear
stress $\langle u^{\prime}w^{\prime}\rangle$ (c) and the wall-normal particle
flux $\langle w^{\prime}\phi^{\prime}\rangle$ (d) for measured data (+) and
$LES$ results (—–).

Figure 3: Vertical profiles of (a) mixing length, (b) Schmidt number defined
as $S_{w\phi}=-\nu^{t}d\langle\phi\rangle/dz/W_{s}^{0}\langle\phi\rangle$ Eq.
(12) (dotted line : empirical model Van Rijn (1984) $S_{w\phi}=0.33$ Eq. 4),
(c) Schmidt number based on the resolved Reynolds flux
$S_{w^{\prime}\phi^{\prime}}=\nu^{t}d\langle\phi\rangle/dz/\langle
w^{\prime}\phi^{\prime}\rangle$ Eq. (13) (d) dimensionless settling velocity:
${\rm C_{w}}=\langle
w^{\prime}\phi^{\prime}\rangle/W_{s}^{0}\langle\phi\rangle$ Eq. (14) (dotted
line : empirical formula Akutina  _et al._ (2020) ${\rm C_{w}}=0.16$ Eq. 14).
Measured data RB15 (black +) and two-fluid LES results (blue —–).

In order to avoid any uncertainties associated with a modification of the von
Kármán constant, the eddy viscosity is calculated directly as:

$\nu^{t}=\frac{|\langle u^{\prime}w^{\prime}\rangle|}{\frac{d\langle
u\rangle}{dz}}.$ (11)

To better illustrate the change in the von Kármán constant value, figure 3a
presents the turbulent mixing length $l_{m}=\sqrt{|\langle
u^{\prime}w^{\prime}\rangle|}/d\langle u\rangle/dz$). The dashed black line in
the figure represents the best-fit of the analytical model
$l_{m}=\kappa(z-z_{d})$ where the slope of the line is the von Kármán
constant. As pointed out in Revil-Baudard  _et al._ (2015), the von Kármán
constant value obtained in the experiment is $\kappa\approx 0.22$, a
significantly lower value than 0.41 expected for clear water flow conditions.
The two-fluid LES reproduces this reduction very well. The modification of the
von Kármán ”constant” value has significant fundamental consequences for
boundary layer modeling in particle-laden flows. Furthermore, as already
mentioned, the von Kármán ”constant” appears at the denominator of the Rouse
number and severely impacts suspended particle transport modeling.

Figure 4: Profile of $z^{+}dU^{+}/dz^{+}$ as a function of $z^{+}$ in semi-
logarithmic scale. Measured data RB15 (black +) and two-fluid LES results
(blue —–) ; the horizontal black dash-dotted line represents $1/0.41$ and the
horizontal red dashed line represents $1/0.22$ ; the vertical black dotted
lines represents the range of the log-layer $3\sqrt{{\rm
Re_{\tau}}}<z^{+}<0.15{\rm Re_{\tau}}$ with ${\rm
Re_{\tau}}=u_{*}H_{f}/\nu^{f}=8,500$.

In order to confirm the existence of the logarithmic layer in this particle-
laden flow configuration, the classical diagnostic for single phase flows is
presented. It consists in plotting the product $z^{+}dU^{+}/dz^{+}$ as a
function of $z^{+}$ (e.g. Smits  _et al._ , 2011). A first difficulty to
overcome in particle-laden flows over erodible beds is the definition of the
origin for $z^{+}$. In this work, we suggest to use $z_{d}$, the virtual
origin of the mixing length:

$z^{+}=\dfrac{(z-z_{d})u_{*}}{\nu^{f}}$

Figure 4 shows the $z^{+}dU^{+}/dz^{+}$ as a function of $z^{+}$ in semi-
logarithmic scale. Both RB15 data and the two-fluid LES exhibit a plateau
confirming the existence of the log-layer. The range of $z^{+}$ over which the
log layer exists does not correspond exactly with the classical values for
single phase flows: $3\sqrt{{\rm Re_{\tau}}}<z^{+}<0.15{\rm Re_{\tau}}$ with
${\rm Re_{\tau}}=u_{*}H_{f}/\nu^{f}=8,500$ (vertical dotted lines in figure
4). This difference may be attributed either to the definition of the origin
of $z^{+}$ or to turbulence modifications induced by the presence of
particles. Nevertheless, the existence of a plateau confirm that the flow
velocity follow a logarithmic profile. The range retained to estimate the von
Kármán constant is $z/H_{f}\in[0.05;0.275]$ corresponding to
$z^{+}\in[0;2\times 10^{3}]$ with our definition of $z^{+}$.

The classical method to estimate the turbulent Schmidt number is indirect and
uses Eq. 1 by assuming a local balance between the gradient-diffusion model
(Eq. (2)) and the gravity-driven settling flux $W_{s}^{0}\langle\phi\rangle$
with the particle settling velocity estimated by its value in quiescent water:

$S_{w\phi}=\frac{\nu^{t}}{W_{s}^{0}\langle\phi\rangle}\Big{|}\frac{d\langle\phi\rangle}{dz}\Big{|}.$
(12)

This quantity will be denoted as $S_{w\phi}$ and is shown in figure 3b. For
$z/H_{f}\gtrsim 0.1$, $S_{w\phi}$ is more or less constant with a value below
unity $S_{w\phi}\approx 0.4$ for RB15 data and $S_{w\phi}\approx 0.7$ for two-
fluid LES results. The model proposed by Van Rijn (1984) (see Eq. 4) gives a
value of ${\rm\overline{S_{c}}}=0.3$ which is very close to the experimental
value.

The direct method is based on the measured or simulated eddy viscosity and
Reynolds particle flux (Eq. 2). Using these quantities, the turbulent Schmidt
number can be calculated without using Eq. 1 as:

$S_{w^{\prime}\phi^{\prime}}=\frac{\nu^{t}}{\langle
w^{\prime}\phi^{\prime}\rangle}\Big{|}\frac{d\langle\phi\rangle}{dz}\Big{|}.$
(13)

This quantity will be denoted as $S_{w^{\prime}\phi^{\prime}}$ and is shown in
figure 3c. The profiles are almost constant for $z/H_{f}\geq 0.1$, and the
measured $S_{w^{\prime}\phi^{\prime}}$ is about 4 in RB15 while the two-fluid
LES confirms a value of $S_{w^{\prime}\phi^{\prime}}$ well above unity, even-
though slightly smaller. As mentioned above, measurement of the Reynolds
particle flux remains a particularly challenging task which, to the best of
the authors knowledge, has only been provided to date in Shen and Lemmin
(1999) and Revil-Baudard  _et al._ (2015) for similar type of particle-laden
open-channel flows. Our analysis demonstrated that the estimation of Schmidt
number is highly dependent on the methodology used.

In a multi-phase flow approach, it is well-known that inertial particles
cannot respond instantaneously to the wide range of fluid velocity
fluctuations that exists in a highly turbulent flow. The particle dynamics may
be significantly different from that of the fluid parcels Balachandar and
Eaton (2010). Particle inertia is estimated using the particle Stokes number
${\rm St_{\eta}}=\tau_{p}/\tau_{\eta}$ where $\tau_{\eta}$ is the Kolmogorov
time-scale associated with the smallest turbulent eddies. Inertial particles
correspond to ${\rm St_{\eta}}>1$ which is the case for the configurations
investigated herein (see table 1). This supports the observation that
turbulent dispersion of particles should be less efficient than turbulent
mixing of fluid momentum i.e. $S_{w^{\prime}\phi^{\prime}}>1$. Therefore, the
indirect estimate of turbulent Schmidt number lower than unity $S_{w\phi}<1$
is hypothesized to be due to reduction of settling velocity in the settling
flux. In order to prove this point, a dimensionless settling velocity ${\rm
C_{w}}$ is deduced from the mass balance equation, as:

${\rm C_{w}}=\frac{\langle w^{\prime}\phi^{\prime}\rangle}{\langle\phi\rangle
W_{s}^{0}}.$ (14)

This quantity is plotted in figure 3d which shows an almost constant value in
the dilute suspended-load layer ($z/H_{f}\geq 0.1$) for both RB15 and two-
fluid LES results. The measured quantity shows that the settling velocity is
about ten times smaller than the one in quiescent water i.e. ${\rm
C_{w}}\approx 0.1$. Recently Akutina  _et al._ (2020), using the same PMMA
particles as the one used in RB15 experiments, have measured a strong settling
retardation for individual particles in a turbulence water column under quasi-
homogeneous and isotropic turbulent flow conditions. In these experiments, the
turbulence was generated by facing pairs of vertical grids oscillating
horizontally. The measured settling velocity of individual particles has been
observed to drastically decrease with increasing turbulence intensity
$w_{rms}$. Using five different particle types covering a wide range of size,
density and settling velocity in quiescent water, the authors have shown that
all the data points for the dimensionless settling velocity ${\rm C_{w}}$ as a
function of the dimensionless turbulent intensity $w_{rms}/W_{s}^{0}$ collapse
on a master curve given by:

${\rm C_{w}}=1-1.3\dfrac{w_{rms}}{W_{s}^{0}},$ (15)

for $0.2<w_{rms}/W_{s}^{0}<0.6$. In the present open-channel flow experiment,
the vertical turbulent intensity can be estimated as $w_{rms}\approx
0.8u_{*}\approx 4$ cm/s and Eq. (15) gives a value of ${\rm C_{w}}=0.16$. This
value is very close to the one shown in figure 3d.

The two-fluid LES also predicts settling retardation (${\rm C_{w}}\approx
0.4$) even though the reduction is less strong than in the experiments.
Despite the discrepancy in magnitude, both numerical and measured values
strongly support the hypothesis that settling retardation is responsible for
the indirectly measured Schmidt number value smaller than unity.

To further support our hypothesis, the measured data reported by Shen and
Lemmin (1999) are further examined using the same methodology as described
above. The hydrodynamic conditions of these two runs (red and green symbols in
Figures 5 and 6) are very similar to RB15 flow conditions (see figure 5 and
table 1), the major difference concerns the particles properties that are
denser and smaller, leading to smaller inertia (see section II.A), with a
suspended particle concentration two orders of magnitude smaller.

Figure 5: Comparison of RB15 data with $SL99\\_1$ (green diamond $\diamond$)
and $SL99\\_2$ (red circle $\circ$) data from Shen and Lemmin (1999) reported
for two different hydrodynamic conditions. Same panels as in figure 2.

Figure 6: Comparison of RB15 data (black +) with $SL99\\_1$ (green diamond
$\diamond$) and $SL99\\_2$ (red circle $\circ$) data from Shen and Lemmin
(1999). Same panels as figure 3. Dashed lines in panel (d) represents linear
fit to the data.

Figure 6a shows the mixing length profiles with a von Kármán constant value
very close to 0.41. This suggests that for lower suspended particle
concentration, no reduction of the von Kármán constant is observed. In figure
6b, the turbulent Schmidt number $S_{w\phi}$ shows a similar trend as for RB15
in the near wall region where the value is lower than unity but the scatter in
Shen and Lemmin (1999) increases away from the wall where $S_{w_{\phi}}$
exceeds unity. The vertical profiles of $S_{w^{\prime}\phi^{\prime}}$ are
shown in figure 6c. Almost all data reveal values well above unity supporting
lower turbulent particle concentration diffusivity compared to eddy-viscosity.
Finally, the vertical profile of the dimensionless settling velocity ${\rm
C_{w}}$ is shown in figure 6d. Both runs exhibit values smaller than unity
confirming the hypothesis of settling retardation but with a weaker magnitude.
Furthermore, the ${\rm C_{w}}$ profiles increase linearly with distance from
the bed whereas the RB15 data exhibit a fairly constant value.

Both experimental datasets Revil-Baudard  _et al._ (2015); Shen and Lemmin
(1999) and the two-fluid LES Cheng  _et al._ (2018) support the conclusion
that the actual turbulent Schmidt number is higher than unity and that
settling retardation plays an important role in sediment transport. In the
following, we will use existing data to strengthen the analysis on settling
retardation in turbulent flow. In figure 7a the dimensionless settling
velocity ${\rm C_{w}}$ is plotted as a function of the local volumetric
concentration in semi-log scale to test the hypothesis of hindrance effects.
The dashed line in the figure represents the classical Richardson and Zaki
(1954) empirical formula ${\rm C_{w}}=(1-\phi)^{4.65}$. Clearly, the observed
reduction of settling velocity can not be explained by hindrance effect.
Indeed settling reduction occurs for volumetric concentration as low as
$10^{-3}$ for Shen and Lemmin (1999) data and there is almost no variation
with $\phi$ in RB15 data. In figure 7b the dimensionless settling velocity
${\rm C_{w}}$ is plotted as a function of $W_{rms}/W_{s}^{0}$ as suggested by
Akutina  _et al._ (2020). As already pointed out, the scaling proposed by
Akutina  _et al._ (2020) is compatible with RB15 data, however experiments by
Shen and Lemmin (1999) corresponds to $W_{rms}/W_{s}^{0}>1$ which are out of
Akutina  _et al._ (2020) experimental range. It would be interesting to test
this range of $W_{rms}/W_{s}^{0}$ in the settling column to conclude on the
observed trends in Shen and Lemmin (1999) data but this is beyond the scope of
the present contribution. In figure 7c, the dimensionless settling velocity
${\rm C_{w}}$ is plotted as a function of ${\rm St_{\eta}}$ where the
Kolmogorov time-scale has been evaluated as
$\tau_{\eta}=(\nu^{f}/\epsilon)^{1/2}$ with $\epsilon=W_{rms}^{3}/d_{p}$. The
choice of $W_{rms}^{3}$ is justified by the anisotropy of the turbulence in
the near-bed region and the focus on the wall-normal particle-flux that is
driven by wall-normal velocity fluctuations. Using this local estimation, the
Stokes number shows that it is lower than unity for Shen and Lemmin (1999) and
higher than unity for Revil-Baudard  _et al._ (2015) data (the data in this
panel are only plotted in the region $0.3>z/H_{f}>0.1$). Therefore, the
observed differences between the two datasets may be attributed to the
differences in particles inertia but this would deserve further investigations
based on experimental data over a wider range of flow conditions and particle
properties.

In order to evaluate the consequences of settling retardation and turbulent
Schmidt number higher than unity on volumetric concentration profile
prediction, Eq. (1) is integrated numerically from $z_{r}/H_{f}=0.1$ to
$z/H_{f}$:

$\displaystyle\langle\phi\rangle(z)=\int_{z_{r}}^{z}\dfrac{{\rm S_{c}}\ {\rm
C_{w}}\ W^{0}_{s}}{\kappa^{2}\ (z-z_{d})^{2}\big{|}{\rm d}\langle
U\rangle/{\rm d}z\big{|}}dz.$ (16)

Using ${\rm S_{c}}=S_{w\phi}$, ${\rm C_{w}}=1$ and $\kappa=0.4$ the result of
Eq. (16) is denoted as Rouse Orig in figure 8 while using ${\rm
S_{c}}=S_{w^{\prime}\phi^{\prime}}$ and measured ${\rm C_{w}}$ (using fit from
figure 6 for SL99_1 and 2) the result of Eq. (16) is denoted as Rouse New in
figure 8. For RB15, the dash dotted line is obtained by using $S_{w\phi}$ and
${\rm C_{w}}=1$ but with $\kappa=0.22$ as deduced from the fit in figure 3.
The original Rouse formulation provides poor predictions of concentration
profiles for all cases. The correction provided by $S_{w\phi}$ fails to
predict the slope of the concentration profiles for SL99_1 and SL99_2. For
RB15 data with high sediment concentration, the modification of the von Kármán
constant has a significant impact on the prediction illustrating the
importance of accounting for turbulence modifications, i.e. modification of
$\kappa$, in the prediction of suspended particle concentration. For all
cases, using a turbulent Schmidt number $S_{w^{\prime}\phi^{\prime}}=3.75$ and
a dimensionless settling velocity ${\rm C_{w}}$ provides the best prediction.
Nevertheless, the rather crude resolution in SL99_1 and 2 data and the limited
range of parameters investigated herein is not sufficient to give definitive
conclusion on the parametrization for the Rouse profile.

Table 3: Summary of Rouse profile parameters. The turbulent Schmidt numbers and dimensionless settling velocity are averaged vertically over the range $0.25>z/H_{f}>0.1$ for all configurations except ${\rm C_{w}}$ for SL99_1 and 2 for which a linear fit has been performed. Name | Reference | $\kappa$ | $z_{d}/H_{f}$ | $S_{w\phi}$ | $S_{w^{\prime}\phi^{\prime}}$ | ${\rm C_{w}}$
---|---|---|---|---|---|---
$RB15$ | Revil-Baudard  _et al._ (2015) | $0.22$ | 0.046 | $0.38$ | $3.75$ | $0.1$
$SL99\\_1$ | Shen and Lemmin (1999) | $0.4$ | 0.052 | $1.06$ | $3.24$ | $4.3\times(z/H_{f}-0.066)$
$SL99\\_2$ | Shen and Lemmin (1999) | $0.4$ | 0.053 | $0.79$ | $3.01$ | $3.3\times(z/H_{f}-0.061)$

Figure 7: Evaluation of different settling retardation mechanisms in RB15 data
(black +) , SL99_1 (red circle $\circ$) and SL99_2 (green diamond $\diamond$)
data. (a) Hindered settling function ${\rm C_{w}}=f(\phi)$, the black dashed
line represents the Richardson and Zaki (1954) function (${\rm
C_{w}}=(1-\phi)^{4.65}$) ; (b) Scaling proposed by Akutina  _et al._ (2020)
${\rm C_{w}}=f(W_{rms}/W^{0}_{s})$ and empirical law model proposed by Akutina
_et al._ (2020) (Eq. 14) ; (c) Scaling with the Stokes number ${\rm
C_{w}}=f({\rm St_{\eta}})$ where ${\rm St_{\eta}}=\tau_{p}/\tau_{\eta}$ with
$\tau_{p}$ the particle response time and
$\tau_{\eta}=(\nu^{f}/\epsilon)^{1/2}$ and $\epsilon=W_{rms}^{3}/d_{p}$.

Figure 8: Comparison of Rouse profile predictions with RB15 data (black +) ,
SL99_1 (red circle $\circ$) and SL99_2 (green diamond $\diamond$) data. Dotted
lines represents the solution of Eq. (3) with $\kappa=0.4$ and the value of
${\rm S_{c}=S_{w\phi}}$ given in table 3 denoted as Rouse Orig. The dash-
dotted line corresponds to the Rouse Orig. formula using $\kappa=0.22$ for
RB15. Dashed lines represents the solution of Eq. (16) with depth-averaged
values given in table 3 and ${\rm S_{c}=S_{w^{\prime}\phi^{\prime}}}$.

##  IV Conclusion

In summary, the results presented herein reveals that the uncertainties on the
value of the turbulent Schmidt number can be explained by a shortcoming in the
methodology, and the real value of the turbulent Schmidt number estimated from
directly measured Reynolds particle flux is larger than unity, ${\rm
S_{c}}\in[3;4]$ and that the settling velocity is significantly reduced ${\rm
C_{w}}\in[0.1;0.9]$ even at very low particle concentration. In the
experiments with ”large” plastic particles, the dimensionless settling
velocity is almost constant in the near bed region, however, for fine sand
particles, the settling velocity is linearly increasing with the distance from
the bed. A comparison with existing models reveals that the observed settling
reduction can not be explained by hindrance effects, while turbulence-particle
interactions are most probably responsible for this behavior. Another
important result presented here concerns the modification of turbulence in the
boundary layer and its consequences on suspended particle profile. In Revil-
Baudard  _et al._ (2015) measurements and two-fluid LES the von Kármán
constant is reduced by a factor two. The consequences of this turbulence
modification on particle concentration profile are huge and definitely needs
to be accounted for in the Rouse profile parametrization.

In conclusion, innovative highly-resolved measurement technique and numerical
simulations allow us to obtain new insight that changes the paradigm of
suspended particle transport in a boundary layer flow. We have demonstrated
that (i) turbulent dispersion of inertial particles is significantly smaller
than that of fluid parcels and does not depend much on particle size and
density and (ii) settling velocity of inertial particles is reduced and this
reduction depends on particle size and density. This suggests that the
particle properties are sensitive parameters in this problem but the details
of the underlying physical mechanisms are still to be elucidated. Beyond the
importance for sediment transport predictions, these findings are relevant to
a wide range of two-phase flows such as avalanches, turbidity currents,
pneumatic transport and particle deposition in material processing.

## Data availability

The data and post-processing script used to write the manuscript are available
on zenodo at the following address
https://zenodo.org/record/5765565#.Ya_L7VPjJHt (see reference Chauchat  _et
al._ (2021)).

###### Acknowledgements.

J. Chauchat is financially supported by the french ANR project SHEET-FLOW
(ANR-18-CE01-0003). D. Hurther is supported by the french DGA-funded ANR
Astrid Maturation project MESURE (ANR-16-ASMA-0005). Z. Cheng and T.-J. Hsu
are supported by U.S. Strategic Environmental Research and Development Program
(SERDP, MR20-1478) and National Science Foundation (OCE-1635151). The
computations presented in this paper were performed using the GENCI
infrastructure under Allocations A0060107567 and A0080107567 and the GRICAD
infrastructure. The authors would like to thank N. Mordant and P. Frey for
their constructive comments on the manuscript.

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[2]

# A new (but very nearly old) proof
of the Rogers–Ramanujan identities

Hjalmar Rosengren Department of Mathematical Sciences
Chalmers University of Technology and University of Gothenburg
SE-412 96 Göteborg, Sweden<EMAIL_ADDRESS>http://www.math.chalmers.se/~hjalmar

###### Abstract.

We present a new proof of the Rogers–Ramanujan identities. Surprisingly, all
its ingredients are available already in Rogers seminal paper from 1894, where
he gave a considerably more complicated proof.

##  1\. Introduction

The fascinating story of the Rogers–Ramanujan identities

$\displaystyle\sum_{k=0}^{\infty}\frac{q^{k^{2}}}{(1-q)(1-q^{2})\dotsm(1-q^{k})}$
$\displaystyle=\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+1})(1-q^{5j+4})},$ (1a)
$\displaystyle\sum_{k=0}^{\infty}\frac{q^{k(k+1)}}{(1-q)(1-q^{2})\dotsm(1-q^{k})}$
$\displaystyle=\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}$ (1b)

has been told many times. They were first proved by Rogers in a brilliant but
forgotten paper from 1894 and rediscovered by Ramanujan some time before 1913.
Hardy wrote about the seven proofs known in the 1930s: “None of them is both
simple and straight-forward, and probably it would be unreasonable to hope for
a proof which is” [H]. Several dozen more proofs have appeared since then,
using diverse methods. We refer to [A2] for a survey of the proofs available
in the 1980s and [AKP, BP, C, F, GIS, P, St, T] for examples of more recent
work. The purpose of the present note is to give a new proof that, although
far from the shortest possible, we find relatively “simple and straight-
forward”. Almost unbelievably, this proof was very nearly found by Rogers. He
knew all its main ingredients and could easily have given it by adding one or
two sentences to [R2].

Among the many gems buried in Rogers’ paper are the identities [R2, §6,
(5)–(8)]

$\displaystyle G(q)+G(-q)$ $\displaystyle=2F(q)G(q^{16}),$ (2a) $\displaystyle
G(q)-G(-q)$ $\displaystyle=2qF(q)H(-q^{4}),$ (2b) $\displaystyle H(q)+H(-q)$
$\displaystyle=2F(q)G(-q^{4}),$ (2c) $\displaystyle H(q)-H(-q)$
$\displaystyle=2q^{3}F(q)H(q^{16}),$ (2d)

where $G(q)$ and $H(q)$ denote the sums in (1a) and (1b), respectively, and

$F(q)=\prod_{j=0}^{\infty}\frac{1-q^{8j+8}}{1-q^{2j+2}}.$

Below, we will give both a version of Rogers’ proof of (2) and another proof
based on work of Andrews [A1]. After deriving (2), Rogers writes somewhat
dismissively

> These four identities …are sufficiently remarkable in themselves to call for
> mention at this point, although they may all be derived from the
> $\theta$-function values of the series $\phi(q)$, $\psi(q)$ [our $G(q)$ and
> $H(q)$] obtained in the last section. […] It is not, however, in these
> identities that the special interest in the series $\phi(q)$ and $\psi(q)$
> lies.

At this point Rogers has already proved the Rogers–Ramanujan identities. He
realizes that (2) are quite easy to prove for the product sides of (1) and
hence perhaps less interesting than they may seem.

Our observation is simply that the identities (2) uniquely determine $G$ and
$H$ up to normalization. In fact, if they are written as

$\displaystyle G(q)$ $\displaystyle=F(q)\left(G(q^{16})+qH(-q^{4})\right),$
(3a) $\displaystyle H(q)$
$\displaystyle=F(q)\left(G(-q^{4})+q^{3}H(q^{16})\right),$ (3b)

they can be used to recursively compute $G$ and $H$ as power series, starting
from the initial values $G(0)=H(0)=1$. Since Rogers has proved that (2) hold
both for the sum sides and the product sides of (1), this gives an independent
proof of the Rogers–Ramanujan identities, which is considerably easier than
the one given earlier in the same paper.

We will present the resulting proof in some detail, but we stress that apart
from the preceding paragraph everything can be found in [A1] and [R2]. I am
very much indebted to George Andrews for pointing this out to me. I would also
like to thank Andrew Sills for the reference to his master’s thesis and for
making it available [S1].

In §3, we explain how some further known Rogers–Ramanujan-type identities
either appear as byproducts of the proof or can be obtained by slight
variations. We only consider identities that are very close to (1), in that
the product sides are essentially the same. It would be interesting to
investigate whether similar proofs can be given for different results, such as
Rogers’ mod $7$ identities. We hope to return to this question in the future.

##  2\. A proof of the Rogers–Ramanujan identities

We will use the standard notation

$(a;q)_{k}=\prod_{j=0}^{k-1}(1-aq^{j}),\qquad k\in\mathbb{Z}_{\geq
0}\cup\\{\infty\\},$

which is extended to negative indices by $(a;q)_{-k}=1/(aq^{-k};q)_{k}$. More
generally,

$(a_{1},\dots,a_{m};q)_{k}=(a_{1};q)_{k}\dotsm(a_{m};q)_{k}.$

We will write

$R(t;q)=\sum_{k=0}^{\infty}\frac{q^{k^{2}}t^{k}}{(q;q)_{k}}$

and

$G(q)=R(1;q),\qquad H(q)=R(q;q).$

The Rogers–Ramanujan identities (1) then take the compact form

$G(q)=\frac{1}{(q,q^{4};q^{5})_{\infty}},\qquad
H(q)=\frac{1}{(q^{2},q^{3};q^{5})_{\infty}}.$ (4)

Finally, we introduce the multiplicative theta function

$\theta(z;q)=(q,z,q/z;q)_{\infty}.$

The only results that we will need are Euler’s identity

$\frac{1}{(z;q)_{\infty}}=\sum_{k=0}^{\infty}\frac{z^{k}}{(q;q)_{k}},\qquad|z|<1,$
(5)

Jacobi’s identity

$\theta(z;q)=\sum_{k=-\infty}^{\infty}(-1)^{k}q^{k(k-1)/2}z^{k}$ (6)

and the less classical summation

$\frac{\theta(1/z;q)}{(tz;q)_{\infty}}=(t;q)_{\infty}\sum_{k=-\infty}^{\infty}\frac{(-1)^{k}q^{k(k-1)/2}}{(t;q)_{k}z^{k}},\qquad
0<|z|<\frac{1}{|t|}.$ (7)

To some readers these identities may look just as complicated as (1), but they
are much easier to prove. For instance, they can all be obtained as special
cases of Ramanujan’s ${}_{1}\psi_{1}$-summation, a simple proof of which was
given by Ismail [I], see also [A1, App. C].

Following Andrews [A1], we will represent the sums in (1) as constant terms.
In general, if

$f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$

(all series in this paper can be considered as formal or analytic according to
taste) and we let $\operatorname{CT}$ denote the constant term of a Laurent
series in $z$, then

$\displaystyle\operatorname{CT}\theta(1/z;Q)f(xz^{N})$
$\displaystyle=\operatorname{CT}\sum_{l=-\infty}^{\infty}(-1)^{l}Q^{l(l-1)/2}z^{-l}\sum_{k=0}^{\infty}a_{k}x^{k}z^{Nk}$
$\displaystyle=\sum_{k=0}^{\infty}(-1)^{Nk}Q^{Nk(Nk-1)/2}a_{k}x^{k}.$ (8)

We apply this with $f(z)$ as Euler’s series (5). In order to get a factor
$q^{k^{2}}$, we must choose $Q=q^{2/N^{2}}$. We will only need the cases $N=1$
and $N=2$, which give the constant term representations

$R(t;q)=\operatorname{CT}\frac{\theta(1/z;q^{2})}{(-qtz;q)_{\infty}}=\operatorname{CT}\frac{\theta(1/z;q^{1/2})}{(q^{1/2}tz^{2};q)_{\infty}}.$

Next, we write

$\displaystyle\frac{\theta(1/z;q^{2})}{(-qtz;q)_{\infty}}$
$\displaystyle=\frac{\theta(1/z;q^{2})}{(-qtz;q^{2})}\cdot\frac{1}{(-q^{2}tz;q^{2})}=\frac{\theta(1/z;q^{2})}{(-q^{2}tz;q^{2})}\cdot\frac{1}{(-qtz;q^{2})},$
$\displaystyle\frac{\theta(1/z;q^{1/2})}{(q^{1/2}tz^{2};q)_{\infty}}$
$\displaystyle=\frac{\theta(1/z;q^{1/2})}{(q^{1/4}t^{1/2}z;q^{1/2})_{\infty}}\cdot\frac{1}{(-q^{1/4}t^{1/2}z;q^{1/2})_{\infty}}.$

and expand the two factors using (7) and (5). This gives the quadratic
transformation formulas

$\displaystyle R(t;q)$
$\displaystyle=(-qt;q^{2})_{\infty}\sum_{k=0}^{\infty}\frac{q^{k(k+1)}t^{k}}{(q^{2},-qt;q^{2})_{k}}$
(9a)
$\displaystyle=(-q^{2}t;q^{2})_{\infty}\sum_{k=0}^{\infty}\frac{q^{k^{2}}t^{k}}{(q^{2},-q^{2}t;q^{2})_{k}}$
(9b)
$\displaystyle=(q^{1/4}t^{1/2};q^{1/2})_{\infty}\sum_{k=0}^{\infty}\frac{q^{k^{2}/4}t^{k/2}}{(q^{1/2},q^{1/4}t^{1/2};q^{1/2})_{k}}.$
(9c)

These identities were found by Rogers [R2, §6, (2)–(4)]. Except for his way of
writing, the original proof is not difficult; we will review it in §4. Watson
independently proved (9b) and (9c) [W].

For $t=1$ and $t=q$, the equations (9) simplify to

$\displaystyle G(q)$
$\displaystyle=(-q;q^{2})_{\infty}\sum_{k=0}^{\infty}\frac{q^{k(k+1)}}{(-q;-q)_{2k}}$
(10a)
$\displaystyle=(-q^{2};q^{2})_{\infty}\sum_{k=0}^{\infty}\frac{q^{k^{2}}}{(q^{4};q^{4})_{k}}$
(10b)
$\displaystyle=(q^{1/4};q^{1/2})_{\infty}\sum_{k=0}^{\infty}\frac{q^{{k^{2}}/4}}{(q^{1/4};q^{1/4})_{2k}},$
(10c) $\displaystyle H(q)$
$\displaystyle=(-q^{2};q^{2})_{\infty}\sum_{k=0}^{\infty}\frac{q^{k(k+2)}}{(q^{4};q^{4})_{k}}$
(10d)
$\displaystyle=(-q;q^{2})_{\infty}\sum_{k=0}^{\infty}\frac{q^{k(k+1)}}{(-q;-q)_{2k+1}}$
(10e)
$\displaystyle=(q^{1/4};q^{1/2})_{\infty}\sum_{k=0}^{\infty}\frac{q^{{k(k+2)}/4}}{(q^{1/4};q^{1/4})_{2k+1}}.$
(10f)

Let us introduce the auxiliary series

$A(q)=\sum_{k=0}^{\infty}\frac{q^{k^{2}}}{(q^{4};q^{4})_{k}},\qquad
B(q)=\sum_{k=0}^{\infty}\frac{q^{k(k+2)}}{(q^{4};q^{4})_{k}}.$ (11)

Then, (10) can be written

$\displaystyle G(-q^{4})$
$\displaystyle=(q^{4};q^{8})_{\infty}\frac{B(q)+B(-q)}{2},$ (12a)
$\displaystyle G(q)$ $\displaystyle=(-q^{2};q^{2})_{\infty}\,A(q),$ (12b)
$\displaystyle G(q^{16})$
$\displaystyle=(q^{4};q^{8})_{\infty}\frac{A(q)+A(-q)}{2},$ (12c)
$\displaystyle H(q)$ $\displaystyle=(-q^{2};q^{2})_{\infty}\,B(q),$ (12d)
$\displaystyle H(-q^{4})$
$\displaystyle=(q^{4};q^{8})_{\infty}\frac{A(q)-A(-q)}{2q},$ (12e)
$\displaystyle H(q^{16})$
$\displaystyle=(q^{4};q^{8})_{\infty}\frac{B(q)-B(-q)}{2q^{3}}.$ (12f)

We may use (12b) and (12d) to eliminate $A$ and $B$ from the remaining
equations. Writing also

$F(q)=\frac{(-q^{2};q^{2})_{\infty}}{(q^{4};q^{8})_{\infty}}=\frac{(q^{8};q^{8})_{\infty}}{(q^{2};q^{2})_{\infty}}$

we arrive at (2) or, equivalently, (3).

Substituting

$F(q)=\sum_{k=0}^{\infty}f_{k}q^{2k},\quad
G(q)=\sum_{k=0}^{\infty}g_{k}q^{k},\quad
H(q)=\sum_{k=0}^{\infty}h_{k}q^{k},\quad$

into (3) gives the system of recursions

$\displaystyle g_{2k}$ $\displaystyle=\sum_{j=0}^{\lfloor
k/8\rfloor}f_{k-8j}g_{j},$ $\displaystyle g_{2k+1}$
$\displaystyle=\sum_{j=0}^{\lfloor k/2\rfloor}(-1)^{j}f_{k-2j}h_{j},$ (13a)
$\displaystyle h_{2k}$ $\displaystyle=\sum_{j=0}^{\lfloor
k/2\rfloor}(-1)^{j}f_{k-2j}g_{j},$ $\displaystyle h_{2k+1}$
$\displaystyle=\sum_{j=0}^{\lfloor(k-1)/8\rfloor}f_{k-8j-1}g_{j}.$ (13b)

These clearly determine $g_{k}$ and $h_{k}$ from the initial values
$g_{0}=h_{0}=1$.

To complete the proof of the Rogers–Ramanujan identities we need to verify
that (2) hold for the product sides

$G(q)=\frac{1}{(q,q^{4};q^{5})_{\infty}},\qquad
H(q)=\frac{1}{(q^{2},q^{3};q^{5})_{\infty}}.$

As Rogers observed, this reduces to classical theta function identities and we
will not give the full details. We first write

$\displaystyle G(q)$
$\displaystyle=\frac{1}{(q,q^{4},q^{6},q^{9};q^{10})_{\infty}}=\frac{(-q,-q^{9};q^{10})_{\infty}}{(q^{4},q^{6};q^{10})_{\infty}(q^{2},q^{18};q^{20})_{\infty}}$
$\displaystyle=\frac{(q^{8},q^{12};q^{20})_{\infty}\,\theta(-q;q^{10})}{(q^{2};q^{2})_{\infty}}$

and similarly

$H(q)=\frac{(q^{4},q^{16};q^{20})_{\infty}\,\theta(-q^{3};q^{10})}{(q^{2};q^{2})_{\infty}}.$

Thus, we are reduced to the problem of computing the even and odd part of a
theta function. In general, Jacobi’s triple product identity (6) implies the
classical identities

$\displaystyle\frac{\theta(z;q)+\theta(-z;q)}{2}$
$\displaystyle=\sum_{k=-\infty}^{\infty}q^{2k\cdot(2k-1)/2}z^{2k}=\theta(-qz^{2};q^{4}),$
$\displaystyle\frac{\theta(z;q)-\theta(-z;q)}{2}$
$\displaystyle=-\sum_{k=-\infty}^{\infty}q^{(2k+1)\cdot
2k/2}z^{2k+1}=-z\,\theta(-q^{3}z^{2};q^{4}).$

It follows that, for instance,

$\frac{G(q)+G(-q)}{2}=\frac{(q^{8},q^{12};q^{20})_{\infty}(-q^{12},-q^{28},q^{40};q^{40})_{\infty}}{(q^{2};q^{2})_{\infty}}.$

Using that

$(-q^{12},-q^{28};q^{40})_{\infty}=\frac{(q^{24},q^{56};q^{80})_{\infty}}{(q^{12},q^{28};q^{40})_{\infty}}=\frac{(q^{16},q^{24};q^{40})_{\infty}}{(q^{12},q^{28};q^{40})_{\infty}(q^{16},q^{64};q^{80})}$

the expression above is easily reduced to

$\frac{G(q)+G(-q)}{2}=\frac{(q^{8};q^{8})_{\infty}}{(q^{2};q^{2})_{\infty}(q^{16},q^{64};q^{80})_{\infty}}=\frac{(q^{8};q^{8})_{\infty}}{(q^{2};q^{2})_{\infty}}\,G(q^{16}).$

The remaining parts of (2) follow similarly. Alternatively, the identities (3)
can be obtained as special cases of

$\theta(z;q)=\theta(-qz^{2};q^{4})-z\,\theta(-q^{3}z^{2};q^{4}).$

Finally, we note that the recursions (13) have a clear combinatorial meaning.
For instance, for the product side, the first identity in (13a) means that the
number of partitions of $2k$ into parts congruent to $\pm 1$ mod $5$ is equal
to the number of pairs $(\lambda,\mu)$, where $\lambda$ is another partition
into parts congruent to $\pm 1$ mod $5$, $\mu$ is a partition into parts not
divisible by $4$ and $8|\lambda|+|\mu|=k$. For the sum side one has a
corresponding statement where “congruent to $\pm 1$ mod $5$” is replaced by
“without repeated or consecutive parts”. It would be interesting to find
combinatorial proofs of these facts. Proving the analogous statements for all
four identities would lead to a new combinatorial proof of the
Rogers–Ramanujan identities.

##  3\. Some additional results

Having proved the Rogers–Ramanujan identities, we can sum the six series in
(10). This was of course clear to Rogers, and leads to the identities given in
Slater’s list [MSZ, S2, Sl] as number 99, 20, 98, 16, 94 and 96, respectively
(as is noted in [S2], 98 is equivalent to 79, and 94 to 17).

Above, we applied (8) to Euler’s series (5). If we instead use the companion
series

$f(z)=(-z;q)_{\infty}=\sum_{k=0}^{\infty}\frac{q^{k(k-1)/2}z^{k}}{(q;q)_{k}}$

we must take $Q=q^{1/N^{2}}$ to get a factor $q^{k^{2}}$. We will only
consider the case $N=1$, which leads to

$R(t;q)=\operatorname{CT}\theta(1/z;q)(qtz;q)_{\infty}.$ (14)

Writing

$\displaystyle\theta(1/z;q)(qtz;q)_{\infty}$
$\displaystyle=\frac{\theta(1/z;q)}{(-qtz;q)_{\infty}}\,(q^{2}t^{2}z^{2};q^{2})_{\infty}$
$\displaystyle=\frac{\theta(1/z;q)}{(q^{1/2}tz;q)_{\infty}}\,(q^{1/2}tz;q^{1/2})_{\infty}=\frac{\theta(1/z;q)}{(q^{3/2}tz;q)_{\infty}}\,(qtz;q^{1/2})_{\infty}$

and expanding each factor gives the transformations

$\displaystyle R(t;q)$
$\displaystyle=(-qt;q)_{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{k}q^{3k^{2}}t^{2k}}{(q^{2};q^{2})_{k}(-qt;q)_{2k}}$
(15a)
$\displaystyle=(q^{1/2}t;q)_{\infty}\sum_{k=0}^{\infty}\frac{q^{k(3k-1)/4}t^{k}}{(q^{1/2};q^{1/2})_{k}(q^{1/2}t;q)_{k}}$
(15b)
$\displaystyle=(q^{3/2}t;q)_{\infty}\sum_{k=0}^{\infty}\frac{q^{k(3k+1)/4}t^{k}}{(q^{1/2};q^{1/2})_{k}(q^{3/2}t;q)_{k}}.$
(15c)

The identity (15a) is the limit case $B\rightarrow\infty$ of [GS, (7.9)] and
(15b) is [GS, (7.17)]. We have not found (15c) in the literature but it can be
deduced from (15b) by a standard contiguity argument. Namely, splitting the
terms in (15c) using

$(q^{3/2}t;q)_{\infty}\frac{t^{k}}{(q^{3/2}t;q)_{k}}=(q^{1/2}t;q)_{\infty}\left(\frac{t^{k}}{(q^{1/2}t;q)_{k}}+q^{k+\frac{1}{2}}\frac{t^{k+1}}{(q^{1/2}t;q)_{k+1}}\right)$

and then replacing $k$ by $k-1$ in the second sum one finds that (15c) equals
(15b).

In the cases $t=1$ and $t=q$, (15) can be written as

$\displaystyle G(q)$
$\displaystyle=(-q;q)_{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{k}q^{3k^{2}}}{(q^{4};q^{4})_{k}(-q;q^{2})_{k}}$
(16a)
$\displaystyle=(q^{1/2};q)_{\infty}\sum_{k=0}^{\infty}\frac{q^{k(3k-1)/4}}{(q^{1/2};q^{1/2})_{k}(q^{1/2};q)_{k}},$
(16b)
$\displaystyle=(q^{1/2};q)_{\infty}\sum_{k=0}^{\infty}\frac{q^{k(3k+1)/4}}{(q^{1/2};q^{1/2})_{k}(q^{1/2};q)_{k+1}}$
(16c) $\displaystyle H(q)$
$\displaystyle=(-q;q)_{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{k}q^{k(3k+2)}}{(q^{4};q^{4})_{k}(-q;q^{2})_{k+1}}$
(16d)
$\displaystyle=(q^{1/2};q)_{\infty}\sum_{k=0}^{\infty}\frac{q^{3k(k+1)/4}}{(q^{1/2};q^{1/2})_{k}(q^{1/2};q)_{k+1}}$
(16e)
$\displaystyle=(q^{1/2};q)_{\infty}\sum_{k=0}^{\infty}\frac{q^{k(3k+5)/4}}{(q^{1/2};q^{1/2})_{k}(q^{1/2};q)_{k+2}}.$
(16f) The summations (16a)–(16e) are, respectively, 19, 46, 62, 97 and 44 on
Slater’s list (19 is equivalent to 100, and 44 to 63). Most of them,
(16a)–(16c) and (16e), were obtained by Rogers [R1, R3], whereas the earliest
reference for (16d) is Ramanujan’s lost notebook [AB, (11.2.7)]. We have not
found (16f) in the literature, but as explained above it follows easily from
(16e).

The reason that Roger skips (16d) is probably that it follows by contiguity
from an identity that he did know (15 and 95 on Slater’s list),

$H(q)=(-q;q)_{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{k}q^{k(3k-2)}}{(q^{4};q^{4})_{k}(-q;q^{2})_{k}},$
(16g)

see [AB, p. 253] for the derivation of (16d) from (16g). Not surprisingly, one
can obtain (16g) by a slight modification of our proof of (16d). We provide
some details since it is another nice example of Andrews’ constant term
method. We first write

$\frac{(-1)^{k}q^{3k(k-2)}}{(q^{4};q^{4})_{k}(-q;q^{2})_{k}}=\frac{(-1)^{k}q^{k(k-1)}}{(q^{2};q^{2})_{k}}\cdot\frac{(-1)^{2k}q^{2k(2k-1)/2}}{(-q;q)_{2k}},$

which leads to a constant term representation of the right-hand side of (16g),

$(-q;q)_{\infty}\operatorname{CT}\sum_{k=0}^{\infty}\frac{(-1)^{k}q^{k(k-1)}z^{2k}}{(q^{2};q^{2})_{k}}\sum_{l=-\infty}^{\infty}\frac{(-1)^{l}q^{l(l-1)/2}}{(-q;q)_{l}z^{l}}\\\
=\operatorname{CT}(z^{2};q^{2})_{\infty}\frac{\theta(1/z;q)}{(-qz;q)_{\infty}}=\operatorname{CT}(1+z)(z;q)_{\infty}\theta(1/z;q).$

In order to apply (14), we use the quasi-periodicity of the theta function in
the form $z\theta(1/z;q)=-\theta(q/z;q)$. Since the constant term does not
change under $z\mapsto qz$, we can write

$\operatorname{CT}(1+z)(z;q)_{\infty}\theta(1/z;q)=\operatorname{CT}(z;q)_{\infty}(\theta(1/z;q)-\theta(q/z;q))\\\
=\operatorname{CT}((z;q)_{\infty}-(qz;q)_{\infty})\theta(1/z;q)=R(q^{-1};q)-R(1;q).$

This can in turn be written

$\sum_{k=0}^{\infty}\frac{q^{k^{2}}(q^{-k}-1)}{(q;q)_{k}}=\sum_{k=1}^{\infty}\frac{q^{k^{2}-k}}{(q;q)_{k-1}}=\sum_{k=0}^{\infty}\frac{q^{k^{2}+k}}{(q;q)_{k}}$

and we have arrived at the left-hand side of (16g).

Different constant-term proofs of the identities (16a) and (16g) were given by
Sills in his [S1]. In his approach they are derived from the summation of the
series (11) rather than from the original Rogers–Ramanujan identities (1).

##  4\. Rogers’ proof of the quadratic transformations

Rogers’ proof of the quadratic transformations (9) is not difficult, but he
writes it in a strange way. He first invokes Lemma IV from his previous paper
[R1]. This seems to be a red herring, since that result is not actually used.
However, he does use the more elementary [R1, Lemma 1] without explanation.

Although the notation was not used by Rogers, we find it instructive to
introduce the $q$-exponential functions

$\displaystyle e_{q}(x)$
$\displaystyle=\frac{1}{(x;q)_{\infty}}=\sum_{k=0}^{\infty}\frac{x^{k}}{(q;q)_{k}},$
$\displaystyle E_{q}(x)$
$\displaystyle=(-x;q)_{\infty}=\sum_{k=0}^{\infty}\frac{q^{k(k-1)/2}x^{k}}{(q;q)_{k}}.$

Roger defines the $q$-derivative as

$(\delta_{x}f)(x)=\frac{f(x)-f(qx)}{x}.$

It is easy to see that

$\delta_{x}e_{q}(tx)=te_{q}(tx)$ (17)

and hence

$f(\delta_{x})e_{q}(tx)=f(t)e_{q}(tx)$

for any formal power series $f$. In particular,

$e_{q}(y\delta_{x})e_{q}(tx)=e_{q}(tx)e_{q}(ty),$ (18)

which is a $q$-analogue of

$\exp\left(y\frac{d}{dx}\right)e^{tx}=\big{(}e^{tx}\big{|}_{x\mapsto
x+y}\big{)}=e^{tx}e^{ty}.$

Identifying the coefficient of $t^{k}$ in (18) gives

$e_{q}(y\delta_{x})\frac{x^{k}}{(q;q)_{k}}=H_{k}(x,y),$ (19)

where $H_{k}$ are defined by the generating function

$e_{q}(tx)e_{q}(ty)=\frac{1}{(tx,ty;q)_{\infty}}=\sum_{k=0}^{\infty}H_{k}(x,y)t^{k}.$
(20)

In [R1], Rogers proves (19) in a more direct way and uses it to derive (18).
In any case, (19) is the only fact from [R1] that Rogers needs for his proof
of (9).

The second $q$-exponential function satisfies

$\delta_{x}E_{q}(tx)=tE_{q}(qtx),$

which gives

$\delta_{x}^{k}E_{q}(tx)=q^{k(k-1)/2}t^{k}E_{q}(q^{k}tx)$

and

$e_{q}(y\delta_{x})E_{q}(tx)=\sum_{k=0}^{\infty}\frac{q^{k(k-1)/2}y^{k}t^{k}}{(q;q)_{k}}\,E_{q}(q^{k}tx)=(-tx;q)_{\infty}\sum_{k=0}^{\infty}\frac{q^{k(k-1)/2}y^{k}t^{k}}{(-tx,q;q)_{k}}.$

On the other hand, (19) gives

$e_{q}(y\delta_{x})E_{q}(tx)=\sum_{k=0}^{\infty}t^{k}q^{k(k-1)/2}H_{k}(x,y).$

Specializing $t=q$ we obtain [R2, §6, Eq. (1)] in the form

$\sum_{k=0}^{\infty}q^{k(k+1)/2}H_{k}(x,y)=(-qx;q)_{\infty}\sum_{k=0}^{\infty}\frac{q^{k(k+1)/2}y^{k}}{(-qx,q;q)_{k}}.$
(21)

By symmetry, one can interchange $x$ and $y$ on the right-hand side.

Roger now observes that the case $y=xq^{1/2}$ of (20) is

$\sum_{k=0}^{\infty}H_{k}(x,xq^{1/2})t^{k}=\frac{1}{(tx;q^{1/2})_{\infty}},$

which implies

$H_{k}(x,xq^{1/2})=\frac{x^{k}}{(q^{1/2};q^{1/2})_{k}}.$

Using this in (21) and replacing $q$ by $q^{2}$ gives

$\sum_{k=0}^{\infty}\frac{q^{k(k+1)}x^{k}}{(q;q)_{k}}=(-q^{2}x;q^{2})_{\infty}\sum_{k=0}^{\infty}\frac{q^{k(k+2)}x^{k}}{(-q^{2}x,q^{2};q^{2})_{k}}.$
(22a) Interchanging the role of $x$ and $y$ gives the alternative expression
$\sum_{k=0}^{\infty}\frac{q^{k(k+1)}x^{k}}{(q;q)_{k}}=(-q^{3}x;q^{2})_{\infty}\sum_{k=0}^{\infty}\frac{q^{k(k+1)}x^{k}}{(-q^{3}x,q^{2};q^{2})_{k}}.$
(22b) Finally, the case $y=-x$ of (20) is
$\sum_{k=0}^{\infty}H_{k}(x,-x)t^{k}=\frac{1}{(t^{2}x^{2};q^{2})_{\infty}},$
which gives $H_{2k}(x,-x)=\frac{x^{2k}}{(q^{2};q^{2})_{k}},\qquad
H_{2k+1}(x)=0.$ Using this in (21) gives
$\sum_{k=0}^{\infty}\frac{q^{2k(2k+1)}x^{2k}}{(q^{2};q^{2})_{k}}=(-qx;q)_{\infty}\sum_{k=0}^{\infty}\frac{(-1)^{k}q^{k(k+1)/2}x^{k}}{(-qx,q;q)_{k}}.$
(22c)

The identities (22) are indeed equivalent to (9).

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