# In-game Residential Home Planning via Visual Context-aware Global Relation
Learning

Lijuan Liu1, Yin Yang2, Yi Yuan1, Tianjia Shao3, He Wang4, Kun Zhou3
Corresponding author

###### Abstract

In this paper, we propose an effective global relation learning algorithm to
recommend an appropriate location of a building unit for in-game customization
of residential home complex. Given a construction layout, we propose a visual
context-aware graph generation network that learns the implicit global
relations among the scene components and infers the location of a new building
unit. The proposed network takes as input the scene graph and the
corresponding top-view depth image. It provides the location recommendations
for a newly-added building units by learning an auto-regressive edge
distribution conditioned on existing scenes. We also introduce a global graph-
image matching loss to enhance the awareness of essential geometry semantics
of the site. Qualitative and quantitative experiments demonstrate that the
recommended location well reflects the implicit spatial rules of components in
the residential estates, and it is instructive and practical to locate the
building units in the 3D scene of the complex construction.

## Introduction

Customized residential complex design becomes a popular element in modern
MMORPG games. This module allows players to virtually create personalized
housing experiences with a comprehensive construction and design interface.
For instance, a player could have a palace-like mansion with a carefully-
shaped garden of pools and a greenhouse. Possessing such luxury housing is
unlikely to be possible for most of us. Yet, it could enhance the feeling of
belongingness and escalate the joyfulness during the gaming. On the downside,
designing a residential housing complex is not “as easy as pie”, which
requires professional expertise and extensive experiences. Our answer to this
dilemma is to resort to machine learning to prompt smart suggestions during
the user interaction, which is similar to smart typing system that predicts
the following word we will input.

Figure 1: A high-level overview of our pipeline: from an input image of in-
game the home customization site, our system extracts a relation graph. With
other learned features, we train a graph generation network to infer the
deployment of new edges. Finally, our system outputs a location prediction
indicates the “suitableness” for the next building unit.

Following this motivation, we introduce an algorithm for location
recommendation, which interactively provides suggestions to players on where
to place new building components etc. A few techniques have been proposed to
suggest a placement of a new component in an indoor scene automatically (Wang
et al. 2018, 2019; Nauata et al. 2020; Wu et al. 2019). They use deep learning
techniques e.g., the FiLM net (Perez et al. 2018), to predict a component
location as an attribute of the new node. However, these approaches do not
directly transfer to the home construction which occurs in an open outdoor
space. Clearly, planning the building construction of a residential complex
involves elements of diverse dimensions and scales. Therefore, searching all
the possible positions on the site is not efficient. Second, items/elements in
indoor scenes are normally associated with well-defined functional
constraints, which can be fully exploited by the network. However, we have
much weaker functional relations among buildings on the construction site.
Certain types of building units are also exclusive – for instance, one cannot
add extra building blocks on the top of a swimming pool, and we name such
locations _forbidden areas_. This type of exclusiveness is not considered in
previous algorithms.

Given a layout of a housing site, we aim to suggest the user a location
directly without any prior knowledge of the building unit to be placed. Our
method is inspired by planIT (Wang et al. 2019) converting the site layout
into a graph. To reduce the search space, we do not iterate all the candidate
locations on the layout. Instead, we infer a possible location of a building
unit through _global_ graph relations. While building units do not have strong
local/neighborhood dependence, in a complex with multiple building units, we
leverage global relations (i.e., graph edges) among all the building units to
facilitate our prediction. For instance, one does not want to have houses to
fully enclose a golf court. In this way, our graph generation network learns
implicit global constraints from the existing graph and understands how to add
new graph edges following such ineffable rules. To account for the exclusive
units/areas, we extract the essential visual clues of the input scene from the
top view of the site image through a convolutional net and fuse them into the
graph generation network. Concretely, we construct two data structures as the
network inputs. One is the top-down rendered scene image (with the detailed
exclusive units labeled area description), and the other is the scene graph.
Our graph generation network takes as input the scene graphs and integrates
the corresponding visual clues learned from the scene image to learn the
global relations of nodes, and mimic constructing new edges. The scene graph
does not contain the building’s visual geometry semantics, nor can it describe
the forbidden area in the scene. To this end, we introduce a global graph-
image feature matching loss to enable the awareness of the scene geometry
during graph generation. The proposed visual context-aware global relation
learning network can precisely describe the geometric and topological
semantics of the input scene. The auto-regressive generative mode within the
network can effectively model the edge distribution from the existing nodes to
the future nodes. Finally, we infer the recommended location for guiding the
placement from the learned edge distribution.

We have qualitatively and quantitatively evaluated our method on a residential
housing dataset collected from a commercial game. The results show that our
method can effectively model the global spatial rules in the layout of
building components. With the extracted visual clues, our network effectively
avoids suggestions in the forbidden areas and collisions with existing
buildings. The perceptual study and the quantitative evaluation results
demonstrate that our generated location maps yield meaningful and instructive
guidance for the players to place new building units.

## Related Works

#### Residential Scene Layout Synthesis.

Residential scene layout synthesis plays an important role in various domains,
such as game designs and architectural layouts. With the emergence of large
scene datasets, more deep learning based models are proposed to address the
layout generation problem. DeepSynth (Wang et al. 2018) and FastSynth
(Ritchie, Wang, and Lin 2019) introduce iterative generation methods to
synthesize new indoor layouts with representing the unstructured input as top
view rendered images. In GRAINS (Li et al. 2019) the input is represented as a
tree structure and a recursive auto-encoder network is introduced to learn and
sample the layout hierarchies. (Zhang et al. 2020) represents the input as
both arrangement matrix and rendered images and generates scenes in an
attribute-matrix form with a generative adversarial network. PlanIT (Wang et
al. 2019) proposes a two-stage method with first generating a layout plan
encoded as a relation graph and then instantiating the plan through an
autoregressive convolutional generator based on the rendered images. In (Zhang
et al. 2019), a stylistic GAN is proposed to model the relationship between
the style distribution and the enhancements for 3D indoor scenes. A novel
evaluation method is also introduced by (Liu 2019) to evaluate the synthesized
3D indoor scenes qualitatively. In addition to the work mentioned above on the
indoor layout generation, a few researchers have also proposed some techniques
working on floor layout design. For example, (Wu et al. 2019) proposes a two-
stage method to iteratively locate rooms and walls given an input boundary
while (Hu et al. 2020) introduces an interactive solution in which users can
specify some constraints during planning. In (Nauata et al. 2020), they
propose a convolutional message passing network named House-GAN that takes as
input a bubble diagram and outputs the house layout with axis-aligned bounding
boxes. Unlike these tasks of indoor layout and floor plan design, our work
focuses on outdoor home planning, and specifically, we aim to suggest
locations for the new buildings.

#### Graph Generation Networks.

Graphs are natural representations of information in many areas, such as
biology, engineering, and social sciences. Traditional techniques, such as
(Bollobás and Béla 2001; Leskovec et al. 2010; Margaritis 2003; Leskovec,
Kleinberg, and Faloutsos 2007), are based on hand-engineered graph priors that
adhere to a pre-decided distribution, thus the learned generative models do
not have enough capacity to represent the graph structures contained in the
observed data. Inspired from recent advances in deep generative models in
computer vision (Wang, She, and Ward 2019; Kingma and Welling 2019; Kobyzev,
Prince, and Brubaker 2019) and natural language processing (Radford et al.
2019; Brown et al. 2020), recent techniques have shifted towards a learning-
based approach and have made significant progress. (Simonovsky and Komodakis
2018) proposes a VAE based graph generation model to learn to translate a
latent continuous vector to a graph that can generate a graph matrix at once.
However, different node ordering would lead to different graph matrices for
the same graph structure, making the learning process difficult. In (Li et al.
2018), a message passing method is introduced to express probabilistic
dependencies between nodes and edges within a graph, but the message is
passing on every single edge, leading to a complex training process. GraphRNN
(You et al. 2018) proposes a hierarchical RNN framework to generate nodes and
edges alternately. It also proposes a BFS node ordering scheme to improve
scalability. To speed up the generative velocity, GRAN (Liu et al. 2019)
employs an efficient framework to generate one block of edge connections
between nodes at a time. Inspired by GRAN (Liu et al. 2019), we propose a
graph generation stream in our framework to learn the edge distribution
between existing scene building units and the new units.

## The Dataset

We collected near 150K residential garden plans designed by players from a
popular online game, which provides a large area of $165~{}grid\times
183~{}grid$ ($1~{}grid=64pixels$) and multiple building units of different
sizes. Many players are novices to home design and landscaping, or they simply
do not want to spend time on it. Some home designs are more like a collection
of random building units. Significant efforts have been devoted to clean up
the dataset. We first rendered all the designs into images and randomly picked
30K out of them. Those images were sent to an annotation team, consisting of
trained professionals. Each image will be labeled with five grades, and a
ResNet50 was trained with those manually annotated labels. Another about 30K
designs that fell into the top three grades were automatically picked out by
the trained model. Afterwards, the annotation team re-assessed machined-graded
designed, and we kept ones labled in the top three grades. After this
processing, our dataset contains about 28K designs, with $276$ building units
per sample on average. There are $381$ different building units in total,
including $280$ infrastructure units (e.g., walls, doors etc.), $101$
architectural units, and one forbidden unit that can be any shape (i.e., the
pool).

### Relation Graph Extraction

We convert the scene into a directed relation graph $\mathcal{G=(V,E)}$. In
this graph, nodes $\mathcal{V}$ denote scene units, which also have a spatial
coordinate. Edges $\mathcal{E\subseteq V\times V}$ represent the spatial
relations between nodes.

#### Edges.

In order to encode the arrangement relations between the components, the
spatial relationship is described with four direction types i.e., front, back,
right, left and four distance types, next_to, adjacent, proximal, distant,
resulting 16 spatial edge types in total. To model the geometric relationship
between units in more details, we also detect six edge alignment attributes,
namely left side, vertical center, right side, top side, horizontal center,
and bottom side. To extract spatial edges for node $v_{i}$, we first raycast
from the four sides of its oriented bounding box on the $xy$ plane, and then
detect intersection with other nodes. For an intersecting node, an edge is
added to the graph from $v_{i}$ to that node if the node is visible from
$v_{i}$ with more than $15\%$ on one side, and the directions are defined in
the coordinate frame of node $v_{i}$. We set the distance label based on the
distance between the two nodes’ oriented bounding boxes: next_to if
$distance=0$, adjacent if $0<distance\leq 30$, proximal if $30<distance\leq
80$ and distant otherwise. The alignment attributes are added if there is an
edge connecting two nodes. For clarity, we only show one edge between two
nodes. In fact, once one edge is detected between two nodes, we will add
another edge between them (opposite direction, same distance).

#### Nodes.

An obvious strategy is to to represent a building unit as a node in the
relation graph. However, as one layout design in our dataset contains about
276 different units (most of them are infrastructural units), doing so leads
to an over-complicated graph. To this end, we simplify the relation graph by
merging multiple infrastructure units to one node. Two units can be merged if
they satisfy all the following conditions: 1) they are in the same category
and have the same orientation; 2) they have the same height and are aligned in
the $x$-axis or the same width and aligned the $y$-axis; 3) they are next to
each other and are completely visible to each other. After merging, the number
of nodes in the graph is reduced to an average of 63 with the primary
information preserved.

#### Attributes.

We assign attribute vectors to each graph node and edge to encode the
geometrical/semantic information of the corresponding scene. Specifically, for
a node ${v_{i}\in\mathcal{V}}$, its attribute vector is defined as
${\widetilde{v_{i}}=[l_{i}^{T},o_{i}^{T}]}$, where $l_{i}\in\mathbb{R}^{|D|}$
is the one hot encoded vector of the label, and $|D|$ is the number of the
unit labels. ${o_{i}}\in\mathbb{R}^{4}$ is the oriented bounding box of the
unit on the $xy$ plane. For an edge ${e_{k}\in\mathcal{E}}$, its attribute
vector is defined as ${\widetilde{e_{k}}=[t_{k}^{T},d_{k}^{T},m_{k}^{T}]}$, in
which $t_{k}\in\mathbb{R}^{16}$ is the one hot encoded vector of the edge type
(16 edge types in total). $t_{k}\in\mathbb{R}^{1}$ is the distance between two
nodes. $m_{k}\in\mathbb{R}^{6}$ is the alignment vector of the edge.

### Top-down View Representation

We convert the 3D residential home design into a 2D layout with a top-down
orthographic depth render. Doing so brings several benefits. First, since the
forbidden area in the design can be in any shapes, it is difficult to
represent it as a node in the graph. Instead, rendering it into a spatial
image can provide detailed shape information to the network. Second, although
the design is in 3D, most building units are arranged in 2D. The top-view
rendering better reveals spatial outline of the design. Following (Wang et al.
2018), this rendering maps a $165~{}grid\times 183~{}grid$ area to a
$512\times 512$ image.

## Our Method

We propose a visual context-aware graph generation model to learn the edge
distribution for the possible building. Our model consists of two streams: one
is a ConvNet that learns detailed semantic information of each unit from the
rendered image; the other stream is a graph generation network that takes as
input the relation graph and fuses the visual clues learned from the ConvNet
and outputs the edge distribution based on the existing graph for the possible
building unit.

Figure 2: An overview of our proposed visual context-aware attentive message
passing for the $r$-th round. This is a toy example with five building units
and four edges to illustrate a single message passing iteration (“large bld”:
“large building”).

### Visual Context Extraction

We extract transformed visual features from the rendered images with a
ConvNet. It is known that low-level features from a ConvNet characterize the
details of local regions, and high-level features represent the global
structural information of the input image. In our framework, we produce the
transformed visual features using FPN based object detector (Lin et al. 2017)
in a multi-stage manner. We crop the visual feature of each building unit from
the feature pyramids $\\{C_{1},C_{2},C_{3},C_{4}\\}$ through the ROIAlignLayer
(He et al. 2017). Each cropped visual feature is then transformed into a fixed
dimensional visual clue through a convolutional block and finally integrated
into the corresponding node features in the graph relation learning network to
make the learning process visual context-aware (Figure 2). The convolutional
block is a Conv-BN-ReLU block with a kernel size $3\times 3$.

### Context-Aware Global Relation Learning

With the extracted visual clues and the relation graph as input, our global
relation learning model outputs the edge distribution between the existing
nodes and the possible node. Inspired by (Liao et al. 2019), we encode the
edges of the relation graph $\mathcal{G=(V,E)}$ with a label weighted
adjacency matrix $A$. For each edge ${(i,j)\in\mathcal{E}}$, $A_{ij}=t_{ij}$,
where $t_{ij}\in T$ is the edge label, and $T$ is the edge type set. Each row
vector $a_{i}\in A$ is interpreted as a connectivity feature of node $v_{i}$
representing connected relations between $v_{i}$ and other nodes in the graph.
We learn an edge distribution $P(a_{|V|+1}|\mathcal{G})$, which samples
connectivity features of relations between the new node and existing nodes in
the graph. In our experiment, we only model the edge distribution from the
previous nodes to the new node, which can easily infer the opposite relations.
In the following, we describe how to learn edge distribution in detail. More
implementation details of the network structure are provided in the
supplementary material.

#### Graph Node Initialization.

We first translate the adjacency matrix $A$ into an one-hot matrix
$\widetilde{A}$ of size $|V|\times(|T|+1)\times|V|$ with
$\widetilde{A}[i,A_{ij},j]=1$. All the node features
$\widetilde{a_{i}}\in\widetilde{A}$ are padded with zeros to the max dimension
of the adjacency matrix in the whole dataset (443 in our dataset). Together
with the node attribute vector $v_{i}$, the node representation is initialized
as:

$h^{0}_{i}=f_{init}(\widetilde{a_{i}},\widetilde{v_{i}};W_{init}),$ (1)

where $f_{init}$ is a stacked 1D convolutional block transforming the raw
connectivity features $\widetilde{a_{i}}$ into the latent embeddings. Then the
following one-layer MLP takes as input the embeddings and node attribute
vector $v_{i}$ and outputs a node representation $h^{0}_{i}\in\mathbb{R}^{I}$,
where $I$ is the dimension of the node representation. For the new node
representation, we set $h^{0}_{|V|+1}=0$ and $h^{0}_{|V|+1}\in\mathbb{R}^{I}$.

#### Edge Masked Attentive Message Propagation.

With node features (including the node representation and the corresponding
visual clues) and associated attribute vectors, stacked edge masked attentive
message propagation blocks propagate the messages and update the node
representations’ state. For the new node, we assume that it is connected to
all existing nodes with an unknown label. At the $r$-th step, we first compute
the visual semantic augmented node representations for all graph nodes:

$h^{r^{\prime}}_{i}=f_{ctx}(h^{r}_{i},c^{r}_{i};W_{ctx}),$ (2)

where $h^{r}_{i}$ is the node representation, and the $c^{r}_{i}$ is the
corresponding cropped visual clue. $f_{ctx}$ is a two-layer MLP with learnable
parameters $W_{ctx}$ to make the output node representations aware of the
corresponding visual clues.

To propagate messages and update node representations, the multi-head
attention mechanism (Veličković et al. 2018) is used to weight different
messages for different nodes:

$\displaystyle m^{rk}_{ij}$
$\displaystyle=f^{rk}_{msg}(h^{r^{\prime}}_{i},h^{r^{\prime}}_{j},\widetilde{e_{k}};W^{rk}_{msg}),$
(3) $\displaystyle ma^{rk}_{ij}$
$\displaystyle=f^{rk}_{att}(h^{r^{\prime}}_{i},h^{r^{\prime}}_{j},\widetilde{e_{k}};W^{rk}_{att}),$
(4) $\displaystyle\text{att}^{rk}_{ij}$
$\displaystyle=\frac{\text{exp}(ma^{rk}_{ij})}{\sum_{l\in\mathcal{N}(i)}(ma^{rk}_{il})},$
(5) $\displaystyle h^{r+1}_{i}$
$\displaystyle=f^{r}_{\text{GRU}}(h^{r}_{i},\|_{k=1}^{k=K}{\textstyle\sum}_{j\in\mathcal{N}(i)}\text{att}^{rk}_{ij}m^{rk}_{ij};w^{rk}_{\text{GRU}}).$
(6)

Here, $K$ indicates that we use $K$ different attention mechanisms to
transform the messages flowing on the edges. In the $k$-th attention
mechanism, we first compute the message $ma^{rk}_{ij}$ for all triplets
$[v_{i},e_{k},v_{j}]$ (where $v_{i}$ and $v_{j}$ are the two nodes of the edge
$e_{k}$) according to Eq. (3). An edge masked self-attention weights on
messages is then obtained (according to Eq. (4) and Eq. (5)) to compute a
linear combination of the messages for each node. Finally, the graph node
representations are updated with the concatenation of the $K$ different
message combinations from $K$ different attention mechanism (according to Eq.
(6)). In our experiments, $f^{rk}_{msg}$ is a two-layer MLP with learnable
parameters $W^{rk}_{msg}$, $f^{rk}_{att}$ is implemented as a single-layer
forward neural network followed with a ReLU nonlinearity. $\mathcal{N}(i)$
indicates the neighboring nodes for each node $i$, and $w^{r}_{\text{GRU}}$
are the learnable parameters for GRU. We show an example of this process in
Figure 2.

#### Edge distribution Modelling.

After $R$ steps of message propagation, we obtain the final node
representations $h^{R}_{i}$ for each node $i$ and compute the raw messages
from the existing graph to the new node
$m^{R}_{i,|V|+1}=[h^{R}_{i},h^{R}_{|V|+1}]$. We model the edge distribution
from existing nodes to the new node $P(a_{|V|+1}|\mathcal{G})$ with a mixture
of categorical model based on the raw messages:

$\displaystyle P(a_{|V|+1}|\mathcal{G})=\sum^{S}_{k=1}\alpha_{s}\prod_{1\leq
j\leq|V|}\theta_{s,j,|V|+1},$ (7)
$\displaystyle\alpha=\text{Softmax}(\sum_{1\leq
j\leq|V|}f_{\alpha}(m^{R}_{j,|V|+1};W_{\alpha})),$ (8)
$\displaystyle\theta=\text{Sigmoid}(f_{\theta}(m^{R}_{j,|V|+1};W_{\theta})),$
(9)

where $S$ is the number of the mixtures in our experiments. $\alpha$ is the
mixed coefficient of $S$ dimension. $\theta$ is the learned edge probabilities
of different mixtures. Both $f_{\alpha}$ and $f_{\theta}$ are implemented as a
two-layer MLP, and $W_{\alpha}$ and $W_{\theta}$ are the learnable parameters.
The mixture of categorical distribution provides an efficient way to capture
dependence in the output distribution due to the latent mixture components.

#### Losses.

To learn the edge distribution from existing nodes to the new node, we define
the objective function as the negative log posterior probability of the
mixture model:

$\mathcal{L}_{o}=-\sum_{z=1}^{Z}\text{log}P(a_{z,|V|+1}|\mathcal{G}_{z}),$
(10)

where $Z$ is the batch size. To encourage the graph generation network to
perceive the global visual semantics, we add global graph-image matching loss
to minimize the matching score for the graph image pair. We first obtain the
two global features ($\widetilde{v_{G}^{r}}$ and $\widetilde{v_{I}^{r}}$) by
averaging the corresponding node features for simplify. The matching score is
defined as a cosine similarity:

$R(G_{z}^{r},I_{z}^{r})=\frac{\widetilde{v_{G}^{r}}^{T}\widetilde{v_{I}^{r}}}{\|\widetilde{v_{G}^{r}}\|\cdot\|\widetilde{v_{I}^{r}}\|}.$
(11)

Similar to (Xu et al. 2018), for a batch of graph-images
$\\{(G_{z},I_{z})\\}_{z=1}^{Z}$, the posterior probability of image $I_{z}$
being matching with graph $G_{z}$ is computed as:

$P(G_{z}^{r}|I_{z}^{r})=\frac{\text{exp}(\gamma
R(G_{z}^{r},I_{z}^{r}))}{\sum_{b=1}^{Z}\text{exp}(\gamma
R(G_{z}^{r},I_{b}^{r}))},$ (12)

and the paired symmetric loss is defined as the negative posterior
probability:

$\mathcal{L}_{m}^{r}=-\sum_{z=1}^{Z}\text{log}P(G_{z}^{r}|I_{z}^{r})-\sum_{z=1}^{Z}\text{log}P(I_{z}^{r}|G_{z}^{r}).$
(13)

Finally, the objective function of our model is:

$\mathcal{L}=\mathcal{L}_{o}+\sum_{r=1}^{R}\mathcal{L}_{m}^{r}.$ (14)

## Implementation Details

In our implementation, we first extract relation graphs and rendered images
from the unstructured sites. For the ConvNet to extract the visual clues for
each component, we implement it based on Detectron2 and choose ResNet50 as the
backbone. The aspect ratio is set as $[0.25,0.5,1.0,2.0,4.0]$. This detection
model is pretrained on our rendered scene images, and we will not change the
parameters in the following training phases. The cropped features are then
transformed into visual clues with the size of $1,024$ based on the
convolution block. For the graph generation network, we first learn the
initial node representation with a size of $512$. Then we update the node
representations $4$ rounds together with the corresponding visual clues based
on the stack of edge masked attentive message propagation blocks. For each
message passing block, we first obtain the messages with the dimension of
$128$ and then concatenate the 4-head attention mechanism output to update the
node representation. We add global graph-image matching loss at every round of
message passing. To model the latent dependencies between edges, we set the
number of mixtures to $10$ in the edge distribution model. During the training
phase, we set the batch size as $32$ and choose the Adam solver for
optimization, with the initial learning rate of $lr=10^{-4}$. The model is
trained on 4 TitanX 2080 GPUs.

Figure 3: The visual examples of the discrete labels for evaluation.

## Experiments

We have systematically tested the proposed method. We choose 22.4K designs as
the training set and the rest 5.6K are used for testing.

#### Visualization.

In order to intuitively evaluate our experimental results, we first convert
the discrete edges from the existing scene graph to the target node for the
testing set and predictions based on introduced above. Since our purpose is to
recommend a location for the possible building unit, we set the default target
unit size as $[24,24]$ when visualizing edges for both the ground truth
testing dataset and the predictions. During the visualization process, for the
edge set $\\{e\\}_{t}$ from the current scene to the component $t$, the
probability of the location the current edge points to is set as
$1/|\\{e\\}_{t}|$. The final heatmap is the sum of all the locations’
probability values implied by the corresponding edges. The heatmap is
normalized to a maximum value of 1. In our experiment for perceptual study, we
only keep the areas with the probability value greater than $0.5$ in the
heatmap and smooth them with a Gaussian kernel ($\text{kernel\\_size}=5$),
which better indicates our recommendation. Several examples are shown in
Figure 3.

Figure 4: Location predictions using different algorithms on our testing
dataset.

#### Comparisons.

We compare our method with planIT (Wang et al. 2019) and FastSynth (Ritchie,
Wang, and Lin 2019). While those two methods are originally designed for
indoor scenes, they are quite relevant to our method. For a fair comparison,
we implement the partial graph completion of planIT and add only one unit for
each scene during the test. For (Ritchie, Wang, and Lin 2019), we implemented
the Object Location and choose the predicted location map indicated by the
ground truth label. The heatmaps generated in our method are visualized from
discrete edges. It is more coarse-grained than a pixel-level prediction.
Therefore, we enlarge the areas of position with a probability value higher
than the mean probability value for a fair comparison. We enlarge the area for
each position with a size of $[24,24]$ and centroid of itself. Then the
heatmap is thresholded and smoothed as the above visualization method. We also
compared our method with two degraded variants. The first variant is Baseline:
we implemented the GRAN model (Liao et al. 2019) with 5 GNN layers and tested
on our dataset. In this model, the relation graph is the only input. The
second variant is LocRec(w/o): we implemented with the proposed model without
the global graph-image matching loss during the training process. We denote
our model as LocRec in the benchmark reports.

#### Quality Metrics.

We provide two types of metrics to evaluate the quality of predicted location
maps. First, we define two criteria to evaluate the visualized heatmaps. For
the ground truth heatmap $ht_{r}$ and the corresponding predicted heatmap
$ht_{p}$, we first calculated the mask $m$ of the intersection of the two non-
zero areas, then the f1_score on both areas and probabilities are calculated
to evaluate the results. The recall and the precision score of the area are
defined as $\text{ar}=\sum m/\sum(ht_{r}>0)$ and $\text{ap}=\sum
m/\sum(ht_{p}>0)$, the final
$\text{f1\\_score}_{area}=2/(\text{ap}^{-1}+\text{ar}^{-1})$. To calculate the
f1_score on probabilities, the recall and the precision score are defined as
$\text{pr}=\sum\min(ht_{r}[m],ht_{t}[m])/\sum ht_{r}$ and
$\text{pp}=\sum\min(ht_{r}[m],ht_{t}[m])/\sum ht_{p}$. The high f1_score on
area indicates that our model can effectively recommend the location for the
new building units, while the high f1_score on probabilities indicate that our
recommend location is compact and has clear guiding significance.

Since we aim at prompting players where to place new units, we also provide a
ranking choice perceptual study to rank the results generated by different
methods. We provided 60 questions and invited 45 participants to rank the
different results in each question. We provide two questions for each
participant, “does this heatmap clearly specify a location?” and “are you
willing to place a building at this location?”. Participants are asked to rank
the results based on their answers. We define five levels to quantify the
results, and participants are not allowed to give the same rank to different
results in the same question. Rank5 represents the best, while Rank1
represents the worst. A detailed questionnaire is provided in the
supplementary material.

#### Experimental Results.

Figure 1 shows a location map predicted by our method, where the heatmap is
directly visualized from the predicted edges. We observe that the results
generated by visualizing edges occupies a relatively large area, but the
central location with high probability is still obvious, which has good
guiding significance. We report more results in Figure 4 (more results in the
supplementary material), where the background is the input scene, and the
corresponding predicted location map is imposed to the background as a
heatmap. To more clearly point out the locations generated by different
methods, we show the results after thresholding and smoothing in the figure.
For the results generated by our method (i.e., LocRec), we also plot the
generated edges. We observe that our model can effectively learn the relations
contained in the scenes, and the edge set predicted from the learned
distribution has a high consistency and seldom point to multiple areas at the
same time, which ensures the stability of our results.

Method | ar | ap | pr | pp | $\text{f1s}_{a}$ | $\text{f1s}_{p}$
---|---|---|---|---|---|---
LocRec | 0.494 | 0.847 | 0.297 | 0.551 | 0.624 | 0.386
LocRec(w/o) | 0.463 | 0.834 | 0.283 | 0.561 | 0.596 | 0.376
Baseline | 0.424 | 0.802 | 0.292 | 0.450 | 0.555 | 0.348
PlanIT | 0.428 | 0.778 | 0.138 | 0.522 | 0.550 | 0.219
FastSynth | 0.352 | 0.779 | 0.0981 | 0.499 | 0.485 | 0.164

Table 1: The quantitative score on the testing dataset for different methods.
$\text{f1s}_{a}$: $\text{f1\\_score}_{area}$; $\text{f1s}_{p}$:
$\text{f1\\_score}_{probability}$.

We find that the results generated by our method can indicate accurate (only
one peak area exists in the heatmap and the area with large probability values
is very compact) and collision avoidance locations, which is instructive and
meaningful to locate the new building units for players. We also find that our
predictions can reasonably avoid the forbidden areas and are harmonious with
the existing scenes. The corresponding quantitative results are shown in Table
1. It can be seen that our generated locations can hit the ground truth
results in the testing dataset in most cases (with
$\text{f1\\_score}_{area}=62.4\%,\text{f1\\_score}_{prob}=38.6\%$), which
shows that our approach can effectively model the sptial rules within the
units in the scene. The results of the perception study (Table 2) also confirm
that the players accept our recommendation results and are willing to place
buildings in such locations in most cases (with $\text{\\#Rank5}=64\%$).

Method | Rank1 | Rank2 | Rank3 | Rank4 | Rank5
---|---|---|---|---|---
LocRec | 0.0063 | 0.086 | 0.0060 | 0.26 | 0.64
LocRec(w/o) | 0.18 | 0.17 | 0.18 | 0.24 | 0.24
Baseline | 0.16 | 0.16 | 0.44 | 0.15 | 0.081
PlanIT | 0.11 | 0.29 | 0.25 | 0.22 | 0.13
FastSynth | 0.55 | 0.17 | 0.12 | 0.079 | 0.078

Table 2: The resulting scores of the perceptual study for different methods.

The visualization results of planIT are given in Figure 4, the quantitative
results and the perception study results are provided in Tables 1 and 2.
Compared with FastSynth, planIT gives a more reasonable location, which
verifies that the constraint of the extrinsic relation graphs is more
conductive to us recommending a reasonable location. Because planIT relies on
the local relationship of the current unit when recommending locations, it is
more difficult to learn the global relations between units in the scene, the
resulting recommendation locations are much worse than our result. From the
perceptual study results, compared to planIT we observe that the players are
more satisfied with the location recommended by LocRec. This is because the
construction site has a large space, the locations and probability values
recommended by planIT can be scattered. Our method is based on global
relations and leads to consistent recommendations. The pr score in Table 1
also reflects this fact.

We also provide the visualization results of our method and its variants in
Figure 4. The corresponding benchmarks are shown in Tables 1 and 2. As one can
see, our baseline model is effective. Compared to FastSynth, which only inputs
visual semantics, an algorithm based on relation graph is more conducive to
use to learn a compact location, even it may appear to conflict with other
building units in the scene. After adding visual clues reasonably, the learned
locations becomes more effective. Since LocRec(w/o) does not integrate the
global graph-image matching loss into the learning process of edge
distribution, the resulting locations cannot effectively avoid the forbidden
areas (e.g., pools). With the local visual clues and the global graph-image
matching loss for the learning process, our full model can effectively capture
the detailed and global structure of the input scene, and resulting in the
best location prediction.

## Conclusion

We propose an effective location recommendation method based on a visual
context-aware graph generation network. This net learns the global relations
between the building units. To integrate the visual clues to the learning
process, a global graph-image matching loss in also designed to enable the
awareness of the scene geometry during the graph generation. The experimental
results show that our method can generate instructive and meaningful locations
to place the possible units. Currently, our work focus on recommending one
location for the next building unit. In practice, it is more convenient to
recommend multiple choices for different units collectively, which clearly
offers more options to the user during the customization. However, more
building units require more flexibility and ambiguities during the learning.
In the future, we plan to investigate possible solutions to solve this
problem. Besides, quantitative measurement of uncertainty during learning is
also worth exploring.

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# Applications of Laguerre transform to solve Schrödinger-type equations and
Differential Equations of order four

Gabriel López Garza
Universidad Autónoma Metropolitana Iztapalapa
Ciudad de México, México

## Abstract

The finite Laguerre transform is applied to solve Differential Equations
Problems of order higher than two and a one-dimensional steady-state
Schrödinger equation, by using elementary Linear Algebra methods.

##  1 Introduction

The theory of finite Sturm-Liouville transform, as studied in [6], has been
applied to solve differential and partial differential equations since the
fifties of the last century. McCully, in particular, developed the theory of
Laguerre transform in [11], where the author after calculating some transforms
formulas, employs the finite Laguerre transform for solving the one-
dimensional nonhomogeneous heat equation as an instance of an application.
Since then, the Laguerre transform has become a useful tool for solving
differential equations and for other applications. Taking into account the
large body of literature related to Laguerre transforms it is fair to suggest
that wherever Laguerre polynomials appear in relation to ordinary and partial
differential equations, it is possible to use the finite Laguerre transform.
The present article could be inscribed in support of such a hypothesis. In
fact, related to this, it can be mentioned the case of the article [1] where
recently, Alhaidari utilizes Laguerre polynomials to solve, among others, some
examples of the Schödinger equation. To this aim, he applies the basic
properties of Laguerre polynomials, as well as the theory of three-terms
recursion sequences for difference equations of order two, as presented in
[10]. Approximations to the solutions of the Schrödinger equation for the
Morse potential case, in terms of the Laguerre polynomials, are known [7],
[14], but the technique introduced in [1] seems to be new. Nevertheless,
Alhaidari never applies explicitly the Laguerre finite transform which, as the
present article shows, simplifies and clarifies the pertinence of such kinds
of techniques. In particular, the relevance of the most important Sturm-
Liouville transform
$\mathcal{T}[L[y_{n}(x)]]=\lambda_{n}\mathcal{T}[y_{n}(x)]$, where $L$ is a
second order differential operator and $\lambda_{n},y_{n}(x)$ are the
eigenvalues and eigenfunctions related to the operator respectively, is
completely hidden in Alhaidari’s work. By adding and subtracting appropriate
terms, the author avoids the use of transforms, which is valid, but such
procedures do not permit the implementation of algorithms efficiently. By
employing the finite Laguerre transform method, such difficulties can be
overcome without great difficulty.

Another application treated in this article deals with the iteration of a
Strum-Liouville operator, which transform is
$\mathcal{T}[L^{2}[y_{n}(x)]]=(\lambda_{n})^{2}\mathcal{T}[y_{n}(x)],$ simply,
but when used in physical applications it gives rise to equations that look
very complicated and difficult to solve, but with finite transform methods it
is not so. In such class of problems may be included the Laguerre type
orthogonal polynomials discovered by Krall [8], [9], which are instances of
eigen-solutions of fourth-order differential equations that do not satisfy
second order differential equations.

The finite transforms method associates with the equations studied in this
article systems of linear equations which in the case of the Schrödinger
equation leads can be reduced to a three-term recursion formula which are
completely classified and known [10]. In the case of equations of order bigger
than two it is not possible to reduce the equations to systems of three-terms,
but it is possible to reduce them two a triangular system which somehow, it is
easy to solve. In both last cases, ordinary differential equations solutions
are solved by elementary linear algebra methods.

In section 2, the mathematical frame of the Sturm-Liouville finite transform
is briefly summarized to set the context in which the Laguerre finite
transform is inscribed. The basic properties of the Laguerre transform are
also exposed in this section. In section 3 are solved the Coulomb and Morse
instances of the Schrödinger equation as well as a Laguerre-type equation of
fourth-order. The literature on Laguerre transforms is immense and, hence, it
is difficult to know if some transform is already known. The criteria followed
in this article is that, if a formula is not in the classical book of Erdélyi
[5] or in [2], or if a Laguerre transform is not in [13], the proof must be
exhibited. In the Appendix section 5, the reader can find the proofs of some
of such Laguerre transforms. That the solution of the fourth-order equation
studied in this article is a linear combination of Laguerre polynomials is
also proved in the Appendix.

##  2 Mathematical setting

###  2.1 Sturm-Liouville Finite Transform

Consider a Suturm-Liouville boundary value problem

$\displaystyle\mathcal{L}[y](x)=\frac{d}{dx}\left(p(x)\frac{d}{dx}y(x)\right)-q(x)y(y)=-\lambda
r(x)y(x)$ (1) $\displaystyle a_{1}y(\alpha)+b_{1}y^{\prime}(\alpha)=0,\qquad
a_{2}y(\beta)+b_{2}y^{\prime}(\beta)=0,$ (2)

when the eigenfunctions are polynomials. Specifically, in this article is
consider the case of Laguerre polynomials which, as the reader may recall, are
solutions of

$\displaystyle\mathcal{L}[y](x)$ $\displaystyle=$
$\displaystyle\frac{d}{dx}\left(x^{\nu+1}e^{-x}\frac{d}{dx}y(x)\right)=-nx^{\nu}e^{-x}y(x),0<x<\infty.$
(3)

And given that $\beta=+\infty$, the boundary condition (2) become

$\displaystyle\lim_{\beta\to+\infty}r(\beta)(a_{2}y(\beta)+b_{2}y^{\prime}(\beta))=0,$
(4)

where $r(x)=x^{\nu}e^{-x}$, accordingly. Solutions of (3) are denoted
$L^{\nu}_{n}(x)$, for $\nu\neq 0$ and, as is usually done in the literature,
$L^{0}_{n}(x)$ is denoted $L_{n}(x)$ simply. The number $\nu$ is called the
order of the corresponding Laguerre polynomials.

Solutions of the equation (1) which satisfy boundary conditions (2), denoted
by $P_{n}(x),\,n\geq 0$. It is well known that suitable differential functions
$f(x)$ (i.e. functions satisfying the respective boundary value conditions and
sufficiently differentiable to satisfy the corresponding differential
equation), have an expansion, called Sturm -Liouville expansion, of the form

$\displaystyle f(x)$ $\displaystyle=$ $\displaystyle
c_{0}P_{0}(x)+c_{1}P_{1}(x)+\cdots=\sum_{m=0}^{\infty}c_{m}P_{m}(x),$ (5)
$\displaystyle\mbox{where }c_{m}$ $\displaystyle=$
$\displaystyle\frac{\int_{\alpha}^{\beta}r(x)f(x)P_{m}(x)dx}{\int_{\alpha}^{\beta}r(x)P_{m}^{2}(x)dx}.$

Of course, assuming $r(x)>0,\alpha\leq x\leq\beta$, to be continuous in the
interval of definition, the integral $\int_{\alpha}^{\beta}r(x)h(x)g(x)dx$
defines an inner product $\langle h,g\rangle_{r}$ in a corresponding space of
integrable functions.

From formula (5) it is possible to establish a correspondence
$f(x)\leftrightarrow\\{c_{m}\\},m\geq 0$, so that the sequence $\\{c_{n}\\}$
is known as the Sturm-Liouville finite transform of $f(x)$, which is usually
denoted by

$\displaystyle\mathcal{T}[f](s)=\\{c_{m}\\}\stackrel{{\scriptstyle
def}}{{=}}c_{0}+c_{1}s+c_{2}s^{2}+\cdots$ (6)

With the purely formal notation defined in (6) we mean the sum and Cauchy
product of sequences, where $s^{n}$ is the sequence

$\displaystyle s^{n}\stackrel{{\scriptstyle
def}}{{=}}\\{\underbrace{0,0,\dots,0}_{n},1,0,0,\dots\\}$ (7)

The sum of sequences $\\{a_{n}\\},\\{b_{n}\\}$ is defined as usual by
$\\{a_{n}\\}+\\{b_{n}\\}\stackrel{{\scriptstyle def}}{{=}}\\{a_{n}+b_{n}\\}$
and the Cauchy product of sequences is defined by the sequence

$\displaystyle\\{a_{n}\\}\\{b_{n}\\}=\\{a_{0}b_{0},a_{0}b_{1}+a_{1}b_{0},a_{0}b_{2}+a_{1}b_{1}+a_{2}b_{0},\dots,\sum_{\tau=0}^{n}a_{\tau}b_{n-\tau},\dots\\}.$
(8)

The sequence $\\{k,0,0,\dots\\}$ is simply written

$k\stackrel{{\scriptstyle def}}{{=}}\\{k,0,0,\dots\\}=k\\{1,0,0,\dots\\},$

as a simplification in the notation currently used, so by (6) it is understood

$\displaystyle\mathcal{T}[f](s)=\\{c_{m}\\}$ $\displaystyle=$ $\displaystyle
c_{0}+c_{1}s+c_{2}s^{2}+\cdots+c_{n}s^{n}+\cdots$ $\displaystyle=$
$\displaystyle\\{c_{0},0,\dots\\}\\{1,0,\dots\\}+\\{c_{1},0,\dots\\}\\{0,1,0,\dots\\}+\cdots$
$\displaystyle+\\{c_{n},0,\dots\\}\\{\underbrace{0,0,\dots,0}_{n},1,0,0,\dots\\}+\cdots.$

Observe that convergence is not germane to this notation, and (6) merely
states that the term $c_{m}$ of the sequence occupies the same place as the
$1$ in the sequence $s^{m}$. Finally, the last step in solving differential
equations via transforms is to find the inverse transform. In the case of
finite transforms, the inverse transform is find simply writing down the
corresponding series in terms of eigenfunctions [6], i. e. series (5) gives
directly the inverse transform of a sequence

$\displaystyle\mathcal{T}^{-1}[\\{c_{n}\\}]=c_{0}P_{0}(x)+c_{1}P_{1}(x)+c_{2}P_{2}(x)+\cdots$
(9)

Of course, convergence matters in this case, but this aspect is well-known for
Laguerre polynomials or at least boundedness of solutions is warranted for the
problems studied in this article (see [1] for boundedness of solutions of the
Schödinger equation). Nevertheless, such expressions depend on the
characteristics of each particular examples which have to be specified.

###  2.2  Laguerre transforms of order $\nu$

The Laguerre polynomials are defined by

$\displaystyle
L_{n}^{\nu}(x)=\sum_{m=0}^{n}\binom{n+\nu}{n-m}\frac{(-x)^{m}}{m!},$

where
$\displaystyle{\binom{\alpha}{\beta}=\frac{\Gamma(\alpha+1)}{\Gamma(\beta+1)\Gamma(\alpha-\beta+1)}.}$
The basic orthogonality relation between the Laguerre polynomials is given by

$\displaystyle\int_{0}^{\infty}e^{-x}x^{\nu}L_{n}^{\nu}(x)L_{m}^{\nu}(x)dx=\frac{\Gamma(n+\nu+1)}{n!}\delta_{nm},$
(10)

where $\delta_{nm}$ is the Kronecker delta symbol. Such relation leads to the
definition of the Laguerre transform of order $\nu$:

$\displaystyle\mathcal{T}^{\nu}[f(x)]=\left\\{\int_{0}^{\infty}e^{-x}x^{\nu}L_{n}^{\nu}(x)f(x)dx\right\\}=\\{c^{\nu}_{n}\\},$
(11)

it must be emphasize that the Laguerre transform is a sequence of numbers in
$\mathbb{C}$. The inverse Laguerre transform is defined by

$\displaystyle
f(x)=\mathcal{(T^{\nu})}^{-1}[\\{c_{n}^{\nu}\\}]=\sum_{k=0}^{\infty}c^{\nu}_{n}L_{n}^{\nu}(x).$
(12)

##  3 Examples of applications

###  3.1 Applications to the Schrödinger equation

In this section is consider the equation

$\displaystyle-\frac{1}{2}\frac{d^{2}}{dr^{2}}\psi(r)-(V(r)+E)\psi(r)=0$ (13)

which in appropriate units ($\hbar$=M=1) is the steady state Schödinger
equation defined in a one dimensional space, where $V(r)$ is a potential
function and $E$ is the energy. Under the change of coordinates (see [1]),
given by $\lambda^{-1}\xi(x)=dx/dr$, where $\lambda\geq 0$ has inverse length
units, equation (13) becomes

$\displaystyle\lambda^{2}\xi^{2}\left[\frac{d^{2}}{dx^{2}}\psi(x)+\frac{1}{\xi}\frac{d\xi}{dx}\frac{d}{dx}\psi(x)-\frac{2}{\lambda^{2}\xi^{2}}W(x)\psi(x)\right]=0,$
(14)

where $W(x)=V(r)-E$. To obtain a Laguerre-type equation the change of
coordinates must satisfy $x(r)\geq 0$ and setting
$\frac{1}{\xi}\frac{d\xi}{dx}=\frac{a}{x}$, leads to $\xi(x)=x^{a}e^{bx}$.

In this way, equation (14) becomes

$\displaystyle\lambda^{2}\xi^{2}\left[\frac{d^{2}}{dx^{2}}\psi(x)+\left(\frac{a}{x}+b\right)\frac{d}{dx}\psi(x)+\left(A_{+}+\frac{A_{-}}{x^{2}}-\frac{A_{0}}{x}\right)\psi(x)\right]=0,$
(15)

where $A_{\pm},A_{0},a,b$ are real parameters determined in terms of $V(r)$
and $E$.

To solve equation (15) it is proposed a solution of the form

$\displaystyle\psi(x)=x^{\alpha}e^{-\beta x}y(x),$ (16)

where $y=\sum_{k=0}^{\infty}c_{k}L_{k}^{\nu}(x)$, and $L_{n}^{\nu}(x)$ are the
Laguerre polynomials of order $\nu$, and $\alpha,\beta,\nu$ are dimensionless
parameters, free for the moment, but to be determined according to the
concrete examples to be solved below.

To solve (15) the use of the finite Laguerre transform is introduced. Many of
the following transforms are known [13] or are obtained by direct calculation
by using Laguerre polynomial properties found in [2] or in [5].

The following formulas (19), (20) and (LABEL:tnu5), are proved in section 5,
since they are not found in any of the items in the bibliography: [2], [13] or
[5].

Theorem 1. Let $\mathcal{T}^{\nu}[f(x)]=\\{c_{n}^{\nu}\\}$, then

$\displaystyle\mathcal{T}^{\nu}\left[\frac{d}{dx}f(x)\right]$ $\displaystyle=$
$\displaystyle\left\\{\sum_{k=0}^{n}c_{k}^{\nu}-\nu\sum_{k=0}^{n}c_{k}^{\nu-1}\right\\}$
(17) $\displaystyle\mathcal{T}^{\nu}\left[\frac{f(x)}{x}\right]$
$\displaystyle=$ $\displaystyle\left\\{\sum_{k=0}^{n}c^{\nu-1}_{k}\right\\},$
(18) $\displaystyle\mathcal{T}^{\nu}\left[xf(x)\right]$ $\displaystyle=$
$\displaystyle\left\\{(2n+\nu+1)c_{n}^{\nu}-(n+1)c_{n+1}^{\nu}-(n+\nu)c_{n-1}^{\nu}\right\\},$
(19) $\displaystyle\mathcal{T}^{\nu}\left[x\frac{d}{dx}f(x)\right]$
$\displaystyle=$ $\displaystyle\\{nc_{n}^{\nu}-(n+1)c_{n+1}^{\nu}\\},$ (20)
$\displaystyle\mathcal{T}^{\nu}\left[x\frac{d^{2}}{dx^{2}}f(x)\right]$
$\displaystyle=$
$\displaystyle\left\\{-(\nu+1)\left(\sum_{k=0}^{n}c_{k}^{\nu}-\nu\sum_{k=0}^{n}c_{k}^{\nu-1}\right)-(n+1)c^{\nu}_{n+1}\right\\}.$

To begin with, the Coulomb problem is solved next.

####  3.1.1 Coulomb problem

Setting the parameters $a=b=0$ follows that $\xi(x)=1$, so that $x=\lambda r$.
The corresponding form of equation (15) for this problem is

$\displaystyle\frac{\lambda^{2}\xi^{2}}{x}\left[x\frac{d^{2}}{dx^{2}}\psi(x)+\left(A_{+}x+\frac{A_{-}}{x}-A_{0}\right)\psi(x)\right]=0,$
(22)

where
$A_{0}=\frac{2Z}{\lambda},A_{-}=-\mathcal{l}(\mathcal{l}+1),A_{+}=\frac{2E}{\lambda^{2}}$
.

To solve equation (22) it is proposed a solution of the form

$\displaystyle\psi(x)=x^{\alpha}e^{-\beta x}y(x),$ (23)

where $y=\sum_{k=0}^{\infty}c_{k}L_{k}^{\nu}(x)$, $L_{n}^{\nu}(x)$ are the
Laguerre polynomials of order $\nu$, and $\alpha,\beta,\nu$ are dimensionless
parameters, free for the moment, but to be determined according to the
concrete examples to be solved below.

By substituting $\psi$ of (23) in (22) the following equation is obtained,
after canceling factors:

$x\frac{d^{2}}{dx^{2}}y(x)+(2\alpha-2\beta
x)\frac{d}{dx}y+(\beta^{2}+A_{+})xy(x)+\\\
+[\alpha^{2}-\alpha+A_{-}]\frac{y(x)}{x}-(A_{0}+2\alpha\beta)y(x)=0.$ (24)

It is possible to find a solution of equation (24) of the form
$\psi(x)=x^{\alpha}e^{-\beta x}y(x)$, where
$y(x)=\sum_{k=0}^{\infty}c_{k}L_{k}^{\nu}(x)$ since $c_{n}^{\nu}$ can be
calculated and hence, solve equation (24), by means of the finite transform
method.

Applying formulas (17) to (LABEL:tnu5) to equation (24) it is possible to
solve the Coulomb problem. The complete transform of equation (24) is

$[-\nu-1+2\alpha-2\beta n+(\beta^{2}+A_{+})(2n+\nu+1)-\\\
-(A_{0}-2\alpha)c_{n}^{\nu}-(\beta^{2}+A_{+})(n+\nu)c_{n-1}^{\nu}-\\\
-[(n+1)(1-2\beta+\beta^{2}+A_{+})]c_{n+1}^{\nu}+\\\
+(\nu+1-2\alpha)\left(-\sum^{n-1}_{k=0}c_{k}^{\nu}+\nu\sum_{k=0}^{n}c_{k}^{\nu-1}\right)+\\\
+(\alpha^{2}-\alpha+A_{-})\sum_{k=0}^{n}c_{k}^{\nu-1}=0.$ (25)

Remark. Note that equation (25) is not a three-terms recursion formula since
it does include terms order $\nu-1$ as well as terms of order $\nu$.
Nevertheless it is possible to solve some equations by restricting some of the
coefficients of formula (25) by considering physical parameters.

The parameters $\alpha$, $\beta$, $\nu$ are chosen in (25) in such a way that
the equation does not contain terms of order $c^{\nu-1}_{k}$ nor terms that
include a summation sign. In this procedure, it is valid to appeal to Theorem
6.1 [10][p. 139 case II], since, as already mentioned, $\alpha,\beta,\nu$ are
free parameters. Then it is valid to choosing

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\alpha^{2}-\alpha+A_{-}$ (26)
$\displaystyle\beta$ $\displaystyle=$ $\displaystyle\frac{1}{2}$ (27)
$\displaystyle\nu$ $\displaystyle=$ $\displaystyle 2\alpha-1.$ (28)

From (26) it is obtain $\displaystyle{\alpha=\frac{1\pm\sqrt{1-4A_{-}}}{2}}$,
hence by (28) $\nu^{2}=1^{2}-4A_{-}$, so that $\alpha=(1\pm\nu)/2$. Therefore,
for Coulomb problem $\alpha=(1\pm\sqrt{1-4A_{-}})/2$,
$\nu=\pm\sqrt{1-4A_{-}}$, and $1/4\geq A_{-}$. Finally, the choice of
parameters

$\displaystyle A_{0}=\frac{2Z}{\lambda},\qquad
A_{-}=-\mathcal{l}(\mathcal{l}+1),\qquad A_{+}=\frac{2E}{\lambda^{2}},$ (29)

leads to

$\displaystyle\alpha=\mathcal{l}+1,\qquad\nu=2\mathcal{l}+1,\qquad$ (30)

where $Z$ is the electric charge and $\mathcal{l}$ is the angular momentum
quantum number.

Equation (25) leads to a three terms recurrence relation as follows. By
substituting (26), (27), and (28) in the transformed equation (25) it is
obtain after simplification

$(n+1)c_{n+1}^{\nu}+\left(\frac{n+A_{0}+\alpha}{\left(A_{+}+\frac{1}{4}\right)}-(2n+\nu+1)\right)c_{n}^{\nu}+(n+\nu)c_{n-1}^{\nu}=0.$
(31)

Next, substitute the values of (29) in (31). After collecting terms and
simplification it follows

$\displaystyle(n+1)c_{n+1}^{\nu}-2\left(\frac{-Z}{4E}\sin\phi+(\left(n+\alpha)\cos\phi\right)\right)c_{n}^{\nu}+(n+2\alpha-1)c_{n-1}^{\nu}=0,$
(32)

where $\displaystyle{\cos\phi=\frac{4(2E)-\lambda^{2}}{4(2E)+\lambda^{2}}}$.
As noticed by Alhaidari [1], equation (32) is related to the Meixner-Pollaczek
three-terms recurrence relation in [10][eq. (9.7.3), p. 213]. Note that (32)
is not identical with the Meixner-Pollaczek formula since the order
$\nu=2\alpha-1$ differs from the order $(\lambda)$ in equation (9.7.3) in
[10]. Consequently, by setting $\lambda=\frac{\nu+1}{2}=\mathcal{l}+1$ a
solution of equation (22) is found by choosing the coefficients $c_{n}^{\nu}$
as

$\displaystyle
c_{n}^{\nu}=P_{n}^{\left(\frac{\nu+1}{2}\right)}(z;\phi)=\frac{\left(\frac{\nu+1}{2}\right)_{n}}{n!}e^{in\phi}{}_{2}F_{1}{\left[\genfrac{.}{.}{0.0pt}{}{-n\mskip
8.0mu\lambda+iz}{2\lambda};1-e^{-2i\phi}\right]},$ (33)

where, ${}_{2}F_{1}[\cdot]$ is a hipergeometric function, $z=-Z/2E$, and
$(k)_{n}$ is the Pochhammer symbol (for definitions and notation see [10][Ch.
1 sec. 1.4], for instance). Therefore, taking into account the relations in
(30) and substituting in (33), the solution of equation (22) is

$\displaystyle\psi(x)=x^{\mathcal{l}+1}e^{-x/2}\sum_{k=0}^{\infty}P_{k}^{\mathcal{l}+1}(z;\phi)\,L_{k}^{2\mathcal{l}+1}(x),$
(34)

which is equivalent to equation (60) in [1].

Remark. Recall that in this paper the Laguerre polynomials $L_{n}^{\nu}(x)$
are not normalized and that $z,\phi$ are fixed parameters in (34).

####  3.1.2 One dimensional Morse oscillator

For this case the values of $a,b$, are $a=1,\;b=0$ so that $\xi(x)=x$. Hence
by substituting $\xi,\xi^{\prime}$ in equation (15) and after simplification
it is obtain

$\displaystyle
x\frac{d^{2}}{dx^{2}}\psi(x)+\frac{d}{dx}\psi(x)+\left(A_{+}x-A_{0}+\frac{A_{-}}{x}\right)\psi(x)=0.$
(35)

Now, again, it is proposed a solution $\psi(x)=x^{\alpha}e^{-\beta x}y(x)$ of
(35), which after substitution gives

$x\frac{d^{2}}{dx^{2}}y(x)+(1+2\alpha-2\beta x)\frac{d}{dx}y(x)+\\\
+\left((\beta^{2}+A_{+})x-(\beta+2\alpha+A_{0})+\frac{\alpha^{2}+A_{-}}{x}\right)y(x)=0.$
(36)

After taking transforms in (36) with formulas (17) to (LABEL:tnu5), follows
the next equation for the Laguerre coefficients of
$y(x)=\sum_{k=0}^{\infty}c_{n}^{\nu}L_{n}^{\nu}(x)$:

$c_{n}^{\nu}[n(A_{+}-2\beta+2\beta^{2})+\beta(-1-2\alpha+\beta+\beta\nu)+2\alpha-\nu(1-A_{+})-A_{0}+A_{+}]+\\\
-c_{n+1}^{\nu}[(n+1)((1-\beta)^{2}+A_{+})]-\\\
-c_{n-1}^{\nu}(\beta^{2}+A_{+})(\nu+n)+\\\
+(-\nu+\alpha(2+\alpha)+A_{-})\sum_{k=0}^{n-1}c_{k}^{\nu}+\\\
+(\nu^{2}-2\nu\alpha+\alpha^{2}+A_{-})\sum_{k=0}^{n}c_{k}^{\nu-1}=0.$ (37)

With the same simplification criterion as in the Coulomb example, by appealing
to Theorem 6.1 [10][p. 139 case II], it is obtained the following system by
equating the coefficients of the sum symbols to zero

$\displaystyle-\nu+2\alpha$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\alpha^{2}+A_{-}$ $\displaystyle=$ $\displaystyle 0,$ (38)

from (38) it follows that $\nu=2\alpha,\alpha^{2}=-A_{-}$, and, moreover, it
is possible to set $\beta=1/2$ too, as in the Coulomb example. On the other
hand the values of $A_{0},A_{-},A_{+}$ given in [1] are

$\displaystyle A_{0}=\frac{-2V_{1}}{\lambda^{2}},\quad
A_{-}=\frac{2E}{\lambda^{2}},\quad A_{+}=\frac{-2V_{2}}{\lambda^{2}}.$ (39)

By substituting the values of $\alpha,\nu,\beta,A_{0},a_{-},A_{+}$ in (37) a
three-term recursion system is obtained:

$(n+1)c_{n+1}^{\nu}-c_{n}^{\nu}\left[2n\frac{A_{+}-\frac{1}{4}}{A_{+}+\frac{1}{4}}-\frac{\frac{1}{2}(1+\nu)+A_{0}}{A_{+}-\frac{1}{4}}+\nu+1\right]+(n+\nu)c_{n-1}^{\nu}=0.$
(40)

Setting

$\cos\phi=\frac{A_{+}-\frac{1}{4}}{A_{+}+\frac{1}{4}}$

equation (40) becomes

$\displaystyle(n+1)c_{n+1}^{\nu}-2c_{n}^{\nu}\left[\left(n+\frac{\nu+1}{2}\right)\cos\phi+\frac{A_{0}}{2\sqrt{A_{+}}}\sin\phi\right]+(n+\nu)c_{n-1}^{\nu}=0.$
(41)

Now, formula (41) can be compared with Meixner-Pollaczek recursion formula in
[10][eq. (9.7.3), p. 213]:

$(n+1)P_{n+1}^{(\omega)}\left(\frac{A_{0}}{2\sqrt{A_{+}}};\phi\right)-\\\
-2\left[\frac{A_{0}}{2\sqrt{A_{+}}}\sin\phi+(n+\omega)\cos\phi\right]P_{n}^{(\omega)}\left(\frac{A_{0}}{2\sqrt{A_{+}}};\phi\right)\\\
+(n+\nu)P_{n-1}^{(\omega)}\left(\frac{A_{0}}{2\sqrt{A_{+}}};\phi\right)=0.$
(42)

Set $\omega=(\nu+1)/2$ then the solution of the one dimensional Morse operator
is

$\psi(x)=x^{\nu/2}e^{-x/2}\sum_{k=0}^{\infty}\frac{(\nu+1)_{k}}{k!}e^{-in\phi}{}_{2}F_{1}{\left[\genfrac{.}{.}{0.0pt}{}{-n\mskip
8.0mu\mskip
8.0mu\frac{\nu+1}{2}+i\frac{A_{0}}{2\sqrt{A_{+}}}}{\nu+1};1-e^{-2i\phi}\right]}L_{k}^{\nu}(x),$
(43)

where $\nu=\frac{2\sqrt{-2E}}{\lambda}$ and
$(\nu+1)_{k}=(\nu+1)(\nu+2)\cdots(\nu+k-1).$ $\square$

Remark. Given the first argument $-n$ in the hypergeometric function
${}_{2}F_{1}[\cdot]$ in (43) it is worth to mention that such function is a
polynomial for each $n$, since $(-n)_{n+1}=0$.

###  3.2 Second example, Laguerre-type equations of order four

The problem (44) shown below has been of interest in the literature [8], [9].
The following Theorem 2, shows that it is possible to solve fourth-order
Laguerre-type equations with finite transform methods as follows. Note that
$L_{n}^{0}(x)$ is simply denoted by $L_{n}(x)$ and similarly
$\mathcal{T}^{0}=\mathcal{T}$.

Theorem 2.The problem,

$\displaystyle\begin{cases}\displaystyle{\frac{d^{2}}{dx^{2}}\left(x^{2}e^{-x}\frac{d^{2}y}{dx^{2}}\right)-\frac{d}{dx}\left(([2R+2]x+2)e^{-x}\frac{dy}{dx}\right)=e^{-x}\lambda_{m}y,}\\\
y(0)=R>0,\quad
y^{\prime}(0)=\frac{\lambda_{m}y(0)}{-2R}=-\lambda_{m}=-m(m+2R+1),\,m\in\mathbb{N},\end{cases}$
(44)

has solutions of the form

$y_{m}(x)=-L_{0}(x)-L_{2}(x)-\cdots-L_{m-1}(x)+(R+m)L_{m}(x),\quad 0\leq
x<\infty$ (45)

where

$L_{n}(x)=\sum_{k=0}^{n}(-1)^{k}\frac{n!}{(n-k)!(k!)^{2}}x^{k},\;n=0,1,\dots$

are the order zero Laguerre polynomials of degree $n$.

The proof of theorem 2 is in section 5.

Remark.  It is worth to mention that solutions obtained by Krall in [8], [9]
for problem (44) are of the form

$\displaystyle R_{n}(x)=\sum_{k=0}^{n}\frac{(-1)^{k}}{(k+1)!}{n\choose
k}[k(R+n+1)+R]x^{k},\;n=0,1,\dots.$ (46)

Such solutions (46) are obtained by the Frobenius method. That the solutions
obtained by Krall are equal to the solution s obtained in this article i. e.,
that $y_{m}(x)=R_{m}(x),\,m=0,1,\dots$ for all $x\in[0,\infty)$ is shown by
induction in the appendix 5. The novelty of the solutions $y_{m}(x)$ of (44)
is the method of obtaining them that is described below.

Method to obtain solutions of (44). Using (3) the differential equation in
(44) may be written as

$\displaystyle\mathcal{L}^{2}[y]-(2R+1)\mathcal{L}[y]+2\frac{dy}{dx}-2\frac{d^{2}y}{dx^{2}}=\lambda_{m}y,$
(47)

where $L$ is defined in identity (3). The Laguerre transforms of
$\mathcal{T}[L[y]]$, $T[L^{2}[y]]$, $\mathcal{T}[\frac{dy}{dx}]$ are known
[11], and are included below for the convenience of the reader:

$\displaystyle\mathcal{T}[\mathcal{L}[y]]$ $\displaystyle=$
$\displaystyle\\{-nc_{n}\\},\mbox{ where
}\mathcal{T}[y]=\\{c_{n}\\},\;n=0,1,2,\dots$ (48)
$\displaystyle\mathcal{T}[\mathcal{L}^{2}[y]]$ $\displaystyle=$
$\displaystyle\\{n^{2}c_{n}\\},$ (49)
$\displaystyle\mathcal{T}\left[\frac{dy}{dx}\right]$ $\displaystyle=$
$\displaystyle\left\\{\sum_{k=0}^{n}c_{k}-y(0)\right\\}.$ (50)

By integration by parts as in [11], or iteration in formula (50), it is
possible to obtain the formula

$\displaystyle\mathcal{T}\left[\frac{d^{2}y}{dx^{2}}\right]$ $\displaystyle=$
$\displaystyle\left\\{\sum_{k=0}^{n}(k+1)c_{n-k}-(n+1)y(0)-y^{\prime}(0)\right\\}$
(51)

where $y^{\prime}(0)=\frac{dy}{dx}\big{|}_{x=0}$. After formulas (48), (49),
(50), and (51) by taking transforms in equation (47), it is obtain

$\\{n(n+2R+1)c_{n}\\}+2\left\\{\sum_{k=0}^{n}c_{k}-y(0)\right\\}-\\\
-2\left\\{\sum_{k=0}^{n}(k+1)c_{n-k}-(n+1)y(0)-y^{\prime}(0)\right\\}=m(m+2R+1)\\{c_{n}\\}$
(52)

where the formula is valid for $\lambda_{m}=m(m+2R+1),$ and $m\in\mathbb{N}$,
as in [9][p. 262]. After simplification in (52) it is obtained

$\left\\{2ny(0)+2y^{\prime}(0)-2\sum_{k=1}^{n}kc_{n-k}\right\\}=\\{(m-n)(2R+n+m+1)c_{n}\\}.$
(53)

Set $y(x)=\sum_{k=0}^{\infty}c_{k}L_{k}(x)$, given that the Laguerre
polynomials satisfy $L_{n}(0)=1$, $L^{\prime}_{n}(0)=-n$ for all $n\geq 0$,
[2, thm.6.3 ], then

$\displaystyle y(0)=\sum_{k=0}^{\infty}c_{k},\quad
y^{\prime}(0)=-\sum_{k=0}^{\infty}kc_{k}$ (54)

It is possible to show that solutions of (44) are polynomials of degree $m$.
After simplification and taking into account that formulas (54) are
consequently finite, equation (53) becomes simply

$\displaystyle\\{(m-n)(2R+n+m+1)c_{n}\\}=-2\left\\{\sum_{k=n+1}^{m}(k-n)c_{k}\right\\},\quad
0\leq n\leq m.$ (55)

Formula (55) and conditions $\displaystyle{y(0)=\sum_{k=0}^{m}c_{k},\quad
y^{\prime}(0)=-\sum_{k=0}^{m}kc_{k}}$ provide systems of upper triangular
equations,

$\displaystyle
c_{1}+\phantom{000000000}\,c_{2}+\phantom{00000000000}\cdots\phantom{00000000}+c_{m}$
$\displaystyle=R$
$\displaystyle(m-1)(2R+m+2)c_{1}+\phantom{00000000}2(c_{2}+\phantom{0000}+2c_{3}+\cdots+(m-1)c_{m})$
$\displaystyle=0$
$\displaystyle(m-2)(2R+m+3)c_{2}+2(c_{3}+2c_{4}+\cdots+(m-2)c_{m})$
$\displaystyle=0$ $\displaystyle\vdots$ $\displaystyle 2(R+m)c_{m-1}+2c_{m}$
$\displaystyle=0,$

which are easily solved: $c_{0}=c_{1}=\cdots=c_{m-1}=-1$, $c_{m}=R+m$.
Therefore for $m=0,1,2,\dots$ it is obtained a family of eigenfunctions
$y_{m}(x),$ which are the solutions (45) of the eigenvalue problem (44).

##  4 Conclusions

The reader may notice that the solutions obtained for the Schrödinger equation
in this paper, are slightly different from the approximations already known
[7], [14] in terms of Laguerre polynomials. Although the solutions obtained in
the present study include a complete series and not just a few terms
approximation, the discrepancy depends on the convergence of the Sturm-
Liuville expansion of the solutions found. The solutions in this article are
also different from the solutions in [1], however, the reader may note that
the polynomials used in [1] are normalized, which is not the case for the
polynomials used here. In addition, the author in [1] obtained, by solving the
Morse oscillator, a second-order difference equation containing terms in
$n^{2}$. Terms of order $n^{2}$ can be obtained also by taking Laguerre
transforms of equation (35) multiplied by $x$. But in doing so, the author of
this article was unable to reduce the obtained system to a system involving
recursive series of only three terms, which is needed essentially in the
method in [1]. Perhaps it can be achieved, using equivalent formulas of
Laguerre transforms, but this goal was no longer pursued since a solution in
terms of Meixner-Polaczek and Laguerre polynomials was obtained, and therefore
the goals of this article were achieved.

##  5 Appendix

In this section are included the proofs of theorem 1 and theorem 2.

Proof of theorem 1. Formula (17) is shown in [13][p. 11-16, 11-17]. Formula
(18) follows from identity in [5][formula (39) p. 192]

$\displaystyle
L_{n}^{\nu}(x)=\sum_{k=0}^{n}\frac{(\nu-\beta)_{k}}{k!}L_{n-k}^{\beta}(x),$

where $(r)_{k}=r(r+1)\cdots(r+k-1)$.

Note that formula (18) is obtained from last formula since for $\nu-\beta=1$,
it follows
$\mathcal{T}^{\nu}\left[\frac{f(x)}{x}\right]=\left\\{\sum_{k=0}^{n}\frac{(1)_{k}}{k!}c_{n-k}^{\nu-1}\right\\}=\left\\{\sum_{k=0}^{n}c_{k}^{\nu-1}\right\\}$,
given that $(1)_{k}=k!$.

The proof of formula (19), follows from formula (ii) of theorem 6.11 in [2]:

$\displaystyle
xL_{n}^{\nu}(x)=(2n+\nu+1)L_{n}^{\nu}-(n+\nu)L_{n-1}^{\nu}(x)-(n+1)L_{n+1}^{\nu}(x).$
(56)

Effectively,

$\mathcal{T}^{\nu}[xf(x)]=\int_{0}^{\infty}e^{-x}x^{\nu}L_{n}^{\nu}(x)xf(x)dx=\\\
=\int_{0}^{\infty}e^{-x}x^{\nu}\left[(2n+\nu+1)L_{n}^{\nu}-(n+\nu)L_{n-1}^{\nu}(x)-(n+1)L_{n+1}^{\nu}(x)\right]f(x)dx=\\\
=(2n+\nu+1)c_{n}^{\nu}-(n+\nu)c_{n-1}^{\nu}-(n+1)c_{n+1}^{\nu}.$

Formula (20) is obtain as follows

$\mathcal{T}^{\nu}\left[x\frac{d}{dx}f(x)\right]=\left\\{\int_{0}^{\infty}e^{-x}x^{\nu}L_{n}^{\nu}(x)x\frac{d}{dx}f(x)dx\right\\}\\\
=\left\\{\int_{0}^{\infty}e^{-x}x^{\nu+1}\left(L_{n}^{\nu+1}(x)-L_{n-1}^{\nu+1}(x)\right)\frac{d}{dx}f(x)dx\right\\},$

where the last integral follows from the identity
$L_{n}^{\nu}(x)=L_{n}^{\nu+1}(x)-L_{n-1}^{\nu+1}(x),$ [2][Theorem 6.11 (i)].
Now, from formula (17)

$\displaystyle\int_{0}^{\infty}e^{-x}x^{\nu+1}L_{n}^{\nu+1}(x)\frac{d}{dx}f(x)dx=c_{n}^{\nu+1}-(\nu+1)\sum_{k=0}^{n}c_{k}^{\nu}+\sum_{k=0}^{n-1}c_{k}^{\nu+1}$
$\displaystyle\int_{0}^{\infty}e^{-x}x^{\nu+1}L_{n-1}^{\nu+1}(x)\frac{d}{dx}f(x)dx=c_{n-1}^{\nu+1}-(\nu+1)\sum_{k=0}^{n-1}c_{k}^{\nu}+\sum_{k=0}^{n-2}c_{k}^{\nu+1}.$

In this way, by substracting the last two transforms

$\displaystyle\mathcal{T}^{\nu}\left[x\frac{d}{dx}f(x)\right]=c_{n}^{\nu+1}-(\nu+1)c_{n}^{\nu}.$

By using the recurrence relation
$c_{n}^{\nu+1}=(n+\nu+1)c_{n}^{\nu}-(n+1)c_{n+1}^{\nu}$, [13][p 11-18,11.195
(a)] it is obtained

$\displaystyle\mathcal{T}^{\nu}\left[x\frac{d}{dx}f(x)\right]=nc_{n}^{\nu}-(n+1)c_{n+1}^{\nu}$

as claimed.

Finally, formula (LABEL:tnu5) follows from the solutions of equation (3),
equation which is equivalent to

$\displaystyle x\frac{d^{2}}{dx^{2}}f(x)+(\nu+1-x)\frac{d}{dx}f(x)=-nf(x),$
(57)

from (57) the next transformation follows

$\displaystyle\mathcal{T}^{\nu}\left[x\frac{d^{2}}{dx^{2}}f(x)+(\nu+1-x)\frac{d}{dx}f(x)\right]=\\{-nc_{n}^{\nu}\\}.$
(58)

Since $\mathcal{T}^{\nu}$ is linear the following relation holds

$\displaystyle\mathcal{T}^{\nu}\left[x\frac{d^{2}}{dx^{2}}f(x)\right]=\\{-nc_{n}^{\nu}\\}-\mathcal{T}^{\nu}\left[(\nu+1)\frac{d}{dx}f(x)\right]+\mathcal{T}^{\nu}\left[x\frac{d}{dx}f(x)\right].$
(59)

So, formula (LABEL:tnu5) follows directly from (59) and formulas (19), and
(20). $\square$

Remark. Observe that if the operator $\mathcal{L}^{\nu}[f(x)]$ is defined as

$\displaystyle\mathcal{L}^{\nu}[f(x)]=e^{x}x^{-\nu}\frac{d}{dx}\left(x^{\nu+1}e^{-x}\frac{d}{dx}y(x)\right),$

then formula (58) can be obtained from the Sturm-Liouville eigenvalue relation

$\displaystyle\mathcal{T}^{\nu}[\mathcal{L}^{\nu}[L_{n}^{\nu}(x)]]=-n\mathcal{T}^{\nu}[L_{n}^{\nu}(x)],$

which is probably the most important Laguerre-type transform for the purposes
of this article.

Proof of Theorem 2. Proof by induction on $m$. For $m=0$,
$y_{0}(x)=R=R_{0}(x)$ is obtained directly from substitution $m=0$ in (45),
taking into account that $L_{0}(x)=1$ for $x\in[0,\infty)$, as well as direct
substitution in (46). Assume that for $m=1,2,\dots,n$, formula
$y_{m}(x)=R_{m}(x)$ holds, it is necessary to show that
$y_{n+1}(x)=R_{n+1}(x)$. It follows from formula (45) that

$y_{n+1}=-L_{0}(x)-\cdots-L_{n-1}(x)-L_{n}(x)+(R+n+1)L_{n+1}(x)\\\
=-L_{0}(x)-\cdots-L_{n-1}(x)-L_{n}(x)+(R+n+1)L_{n+1}(x)+\\\
+(R+n)L_{n}(x)-(R+n)L_{n}(x).$ (60)

By substituting the inductive hypothesis in (60) it follows that

$\displaystyle y_{n+1}(x)=R_{n}(x)+(R+n+1)(L_{n+1}(x)-L_{n}(x))$ (61)

But

$L_{n+1}(x)-L_{n}(x)=\sum_{k=0}^{n+1}\frac{(-1)^{k}(n+1)!}{(k!)^{2}(n+1-k)!}x^{k}-\sum_{k=0}^{n}\frac{(-1)^{k}n!}{(k!)^{2}(n-k)!}x^{k}\\\
=\sum_{k=0}^{n+1}\frac{(-1)^{k}n!}{k!(k-1)!(n-k+1)!}x^{k}$

So that,

$y_{n+1}(x)=R_{n}(x)+(R+n+1)\sum_{k=0}^{n+1}\frac{(-1)^{k}n!}{k!(k-1)!(n-k+1)!}x^{k}\\\
=R_{n}(x)+(R+n+1)\sum_{k=0}^{n+1}\frac{(-1)^{k}n!(k+1)}{(k+1)!(k-1)!(n-k+1)!}x^{k}\\\
=R_{n}(x)+\sum_{k=0}^{n+1}\frac{(-1)^{k}}{(k+1)!}\frac{n![k(R+n+1)+R+n+1]}{(k-1)!(n-k+1)!}x^{k}\\\
=R_{n}(x)+\sum_{k=0}^{n+1}\left[\frac{n!(k(R+n+1)+R)}{(k-1)!(n-k+1)!}+\frac{(n+1)!k}{(n-k+1)!k!}\right]x^{k}\\\
=R_{n}(x)+\sum_{k=0}^{n+1}\frac{(-1)^{k}}{(k+1)!}\left[{n\choose
k-1}(k(R+n+1)+R)+{n+1\choose k}k\right]x^{k}.$

By using the formula for $R_{n}(x)$ and the Pascal rule
$\displaystyle{{n+1\choose k}={n\choose k}+{n\choose k-1}}$ it follows that

$y_{n+1}(x)=\sum_{k=0}^{n}\frac{(-1)^{k}}{(k+1)!}{n\choose
k}(k(R+n+1)+R)x^{k}+\\\ +\sum_{k=0}^{n}\frac{(-1)^{k}}{(k+1)!}{n\choose
k-1}(k(R+n+1)+R)x^{k}+\\\ +\sum_{k=0}^{n+1}\frac{(-1)^{k}}{(k+1)!}{n+1\choose
k}kx^{k},$

consequently,

$y_{n+1}(x)=\sum_{k=0}^{n}\frac{(-1)^{k}}{(k+1)!}{n+1\choose
k}(k(R+n+1)+R)x^{k}+\sum_{k=0}^{n+1}\frac{(-1)^{k}}{(k+1)!}{n+1\choose
k}kx^{k}\\\ =R_{n+1}(x).$

Therefore $y_{m}(x)=R_{m}(x)$ for all $m\in\mathbb{N}$ and $x\in[0,\infty)$.
$\square$

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