How to Produce Specific Mathematical Representations of Letters

Question

Can anyone please explain how I should produce these symbols in TeX?

Answer 1

I suggest you load the mathalfa package with the options cal=mt and scr=kp. The former option assigns the Math Time Pro calligraphic font face to \mathcal -- note, though, that the mtpro2 package isn't free of charge -- and the KP fonts script font face to \mathscr.

Do check out pp. 8 and 9 of the user guide of the mathalfa package for other possible choices for \mathscr.

The match between the first letter in the OP's screenshot and \mathcal{B} would appear to be perfect. The second letter in the OP's screenshot seems to be halfway between \mathscr{B} and \bm{\mathscr{B}.

\documentclass{article}
\usepackage[cal=mt,scr=kp]{mathalpha}
\usepackage{bm}
\begin{document}
$\mathcal{B} \quad \mathscr{B} \quad \bm{\mathscr{B}}$
\end{document}

Answer 2

Taking like an example the answer of @Philippe Goutet, where where I have made some small changes, considering also the answer of @Mico, you can define your B calligraphic letters:

\documentclass[a4paper,12pt]{article}
\usepackage{mathtools}
\usepackage[cal=mt]{mathalfa}
\DeclareSymbolFont{mathscrUC}{U}{rsfs}{m}{n}
%%%% from the answer and http://ctan.epst-tlemcen.dz/tex-archive/fonts/txfonts/input/txfonts.sty
\makeatletter
\DeclareRobustCommand*{\mathscr}[1]{\gdef\F@ntPrefix{mathscr@char@}%
  @EachCharacter #1@EndEachCharacter}
\long\def\DoLongFutureLet #1#2#3#4{% 
   \def@FutureLetDecide{#1#2@FutureLetToken
      \def@FutureLetNext{#3}\else
      \def@FutureLetNext{#4}\fi@FutureLetNext}
   \futurelet@FutureLetToken@FutureLetDecide}
\def\DoFutureLet #1#2#3#4{\DoLongFutureLet{#1}{#2}{#3}{#4}}
\def@EachCharacter{\DoFutureLet{\ifx}{@EndEachCharacter}%
   {@EachCharacterDone}{@PickUpTheCharacter}}
\def\m@keCharacter#1{\csname\F@ntPrefix#1\endcsname}
\def@PickUpTheCharacter#1{\m@keCharacter{#1}@EachCharacter}
\def@EachCharacterDone @EndEachCharacter{}
% It has been declared only the letter B. 
\DeclareMathSymbol{\mathscr@char@B}{\mathord}{mathscrUC}{`B}
\makeatother
\begin{document}
$\mathcal{B}, \mathscr{B}$
\end{document}