Now I'm studying the topic that uses the special functions frequently, so I find myself in need for some good reference book on the properties and equalities of the special functions. The optimal one might be the book with dictionary-like or handbook type lists of the properties of the special functions with sufficient amount of proofs. There are many books about the special functions but most of them are lack of proofs, or too pedagogically descriptive textbooks. I would rather prefer the book that I can search up the identities fast and can understand why does the equality hold at least briefly. Is there any good book that meets this condition?
As OP stated he was a physics student in a comment, there are a lot of books called (something similar) to "Mathematical Methods for Physicists", which have relevant references, details and techniques for physicists. I'd probably start with the first two.
Arfken, "Mathematical Methods for Physicists"
Boas, "Mathematical Methods in the Physical Sciences "
Courant and Hilbert, "Methods of Mathematical Physics"
Shankar, "Basic Training in Mathematics: A Fitness Program for Science Students" (probably too basic)
Abramowitz and Stegun, Handbook of Mathematical Functions (not proofs, but there are a lot of useful things there, which you can look up elsewhere)
(Website) Wolfram Mathworld
And for PDE's, you can look at specific PDE books:
Zauderer, Partial Differential Equations of Applied Mathematics
One of the many basic linear PDE books, e.g. Haberman's Applied PDE
You may consider looking at Oldham, Myland, and Spanier's book An Atlas of Functions which can be purchased used quite cheaply on Amazon. I would dare suggest it is a beautiful book, with very nice graphs on glossy pages. It is large and it is organized by classes of functions. There aren't many proofs in the book, so I'm not sure if that would deter you from this book.
Another route to go is to get Schaum's Outline of Mathematical Formulas, though it may not be detailed enough for you.
One resource I haven't seen mentioned is the trusty Table of Integrals, Series, and Products by Gradshteyn & Ryhzik. While known primarily as an integral table, this book has a ton of other material including good coverage of special functions.
Further, I second (or third, etc) the suggestion of the Handbook of Mathematical Functions by Abramowitz & Stegun. It is amusing to read a book with numerical tables, however.
I personally prefer online resources:
They are much more convenient to use as compared to books.
three-volume set by Prudnikov-Brychkov-Marychev (for more special stuff).
Finally, for the proofs, I would recommend