Can a multiclass Beguiler/Wizard cast Wizard spells using Beguiler spell slots?

Question

A Wizard can:

To learn, prepare, or cast a spell, the wizard must have an Intelligence score equal to at least 10 + the spell level. The Difficulty Class for a saving throw against a wizard's spell is 10 + the spell level + the wizard's Intelligence modifier.

[...]

Unlike a bard or sorcerer, a wizard may know any number of spells.

According to another question, a Wizard does not need to be able to cast a spell to copy it to their spellbook:

There are no restrictions on spell level aside from the usual checks required to copy a spell into the spellbook.

A Beguiler has the ability to:

A beguiler casts arcane spells, which are drawn from the beguiler spell list. When you gain access to a new level of spells, you automatically know all the spells for that level on the beguiler's spell list. You can cast any spell you know without preparing it ahead of time. Essentially, your spell list is the same as your spells known list. You also have the option of adding to your existing spell list through your advanced learning class feature (see below) as you increase in level.

This could be read to only allow the Beguiler spell list to be expanded with the Advanced Learning class feature, but there are numerous other recognized ways to add spells to spell lists - e.g. the Arcane Disciple feat or the Rainbow Servant prestige class - so Beguilers being able to cast other spells than those on the Beguiler spell list or selected with Advanced Learning seems clear.

And finally, the rules for multiclass characters (PHB, p59) state:

Spells: The character gains spells from all of his or her spellcasting classes. Thus, an experienced ranger/druid may have access to the spell protection from elements both as a ranger and as a druid. Since the spell’s effect is based on the class level of the caster, the player must keep track of whether the character is preparing and casting protection from elements as a ranger or as a druid.

So, can a Beguiler with a 1-level dip in Wizard who has learned Wizard/Sorcerer spells from scrolls cast them using Beguiler spell slots?

Or in other words, does a character have a single pool of arcane spells known, or one for each base casting class?

Answer 1

No

Your reasoning here is unsound:

A beguiler casts arcane spells, which are drawn from the beguiler spell list. When you gain access to a new level of spells, you automatically know all the spells for that level on the beguiler's spell list. You can cast any spell you know without preparing it ahead of time. Essentially, your spell list is the same as your spells known list. You also have the option of adding to your existing spell list through your advanced learning class feature (see below) as you increase in level.

This could be read to only allow the Beguiler spell list to be expanded with the Advanced Learning class feature, but there are numerous other recognized ways to add spells to spell lists - e.g. the Arcane Disciple feat or the Rainbow Servant prestige class - so Beguilers being able to cast other spells than those on the Beguiler spell list or selected with Advanced Learning seems clear.

Yes, it is possible for a beguiler to cast spells other than from those learned from the beguiler class features “spells” and “advanced learning”—but the quoted description from Player’s Handbook II doesn’t consider that possibility. Per that description, beguiler spells only come from the spells and advanced learning features.

The other options you consider—Arcane Disciple, rainbow servant—are exceptions to the quoted Player’s Handbook II rule that beguiler spells come from their own spells and advanced learning class features. The game is an exception-based ruleset, which is why the rules for melee attacks can say “Your attack bonus with a melee weapon is: Base attack bonus + Strength modifier + size modifier,” without any mention of the possibility that you might have Weapon Finesse—Weapon Finesse is an exception to this rule, which means it supercedes the general rule that attack rolls with melee weapons use Strength.

In just the same way, it is possible for Arcane Disciple or rainbow servant to allow a beguiler to cast spells beyond those offered by their own class features, because they are exceptions to that general rule—which both of them spell out:

Add the chosen domain’s spells to your class list of arcane spells. If you have arcane spellcasting ability from more than one class, you must pick which arcane spellcasting ability this feat applies to. Once chosen, this decision cannot be changed for that feat.

^{(Arcane Disciple feat description, Complete Divine pg. 79)}

Cleric Spell Access: A 10th-level rainbow servant can learn and cast spells from the cleric list, even if they don’t appear on the lists of any spellcasting class he has.

^{(Rainbow servant prestige class description, Complete Divine pg. 56)}

Arcane Disciple specifically adds the spells to your class list, allowing those spells to interact with the beguiler spells feature in the usual way, but extended beyond the usual list. Rainbow servant explicitly allows you to learn and cast cleric spells even if your abilities wouldn’t otherwise allow you to, providing an exception to the beguiler rule that you can only learn spells via the spells and advanced learning features. (That said, it’s not clear how a beguiler would learn these spells—see our dedicated Q&A on the subject.)

Wizard, by contrast, does not make itself an exception—it is describing what you can do with your wizard spellcasting, not changing anything about your beguiler spellcasting.

You need something that says that your beguiler spells ability can prepare or otherwise cast spells from your wizard spellbook, and you don’t have anything that says that. To the best of my knowledge, nothing says that.